6%2d6 systems of inequalities - waynesville r-vi school …€¦ ·  · 2014-08-27y x í 7 y x + 7...

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Solve each system of inequalities by graphing. x y x í 3 62/87,21 *UDSK HDFK LQHTXDOLW\ The graph of x LV VROLG DQG LV LQFOXGHG LQ WKH JUDSK RI WKH VROXWLRQ The graph of y x í 3 is also solid and is included in the graph of the solution. 7KH VROXWLRQ RI WKH V\VWHP LV WKH VHW RI RUGHUHG SDLUV LQ WKH LQWHUVHFWLRQ RI WKH JUDSKV RI x DQG y x í This region is shaded in the graph below. y > í2 eSolutions Manual - Powered by Cognero Page 1 6 - 6 Systems of Inequalities

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Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 1

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 2

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 3

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 4

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 5

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 6

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 7

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 8

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 9

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 10

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 11

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 12

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 13

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 14

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 15

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 16

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 17

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 18

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 19

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 20

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 21

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 22

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 23

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 24

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 25

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 26

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 27

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 28

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 29

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 30

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 31

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 32

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 33

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 34

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 35

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 36

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 37

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 38

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 39

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 40

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 41

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 42

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 43

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy?

eSolutions Manual - Powered by Cognero Page 44

6-6 Systems of Inequalities

Solve each system of inequalities by graphing.���x ���

y ��x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x�����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ���x í 3 is also solid and is included in the graph of the solution.

�7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�LQ�WKH�LQWHUVHFWLRQ�RI�WKH�JUDSKV�RI�x�����DQG�y ���x í ���

This region is shaded in the graph below.

���y > í2 y ��x + 9

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� � The graph of y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y ��x ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2 and y ��x + 9.

� �7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

���y < 3x + 8 y ���x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

� The graph of y ����x�LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ�� �

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 3x + 8 and y ����x. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

���3x í y �� í1 2x + y ����

62/87,21���Graph each inequality. The graph of 3x í y �� í1 is solid and is included in the graph of the solution. The graph of 2x + y �����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�VHW�RI�RUGHUHG�SDLUV�in the intersection of the graphs of 3x í y �� í1 and 2x + y ������7KH�VROXWLRQ�UHJLRQ�LV�WKH�GDUNHVW�VKDGHG�DUHD�LQ�WKH�graph below.

���y ���x í 7 y ���x + 7

62/87,21���Graph each inequality. The graph of y ����x í 7 is solid. The graph of y ����x + 7is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 7 and y ����x + 7. This region is shadedin the graph below. Because the graphs do not intersect, there is no solution.

���y > í2x + 5 y � í2x + 10

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > í2x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y �� í2x������LV�VROLG��

Because the lines are parallel and the solution of the system must satisfy y > í2x + 5 and y �� í2x + 10, the line y �� í2x������LV�LQFOXGHG�LQ�WKH�VROXWLRQ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution of the system is the set of ordered pairs in the intersection of the graphs of y > í2x + 5 and y �� í2x + 10. The solution region is shaded in the graph below.

���2x + y ���� 2x + y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x + y �����LV�VROLG��

The graph of 2x + y �����LV�DOVR�VROLG��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x + y �����DQG��x + y ������7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��

� Because the lines are parallel and the solution of the system must satisfy 2x + y �����DQG��x + y ������WKH�OLQH��x + y ���5 is included in the solution but the line 2x + y �����LV�QRW�

���5x í y < í2 5x í y > 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\���7KH�JUDSK�RI��x í y < í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�graph of 5x í y �!���LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ���7KH�VROXWLRQ�RI�WKH�V\VWHP�LV�WKH�set of ordered pairs in the intersection of the graphs of 5x í y < í2 and 5x í y > 6. To find the solution, overlay the graphs and locate the green region. Because the graphs do not intersect, there is no solution.

� �

���$872�5$&,1*� At a race car driving school, there are safety requirements. �

� D�� Define the variables, and write a system of inequalities to represent the height and weight requirements in this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (50, 180) a solution? Explain.

62/87,21���D�� Let h = the height of the driver in inches and w = the weight of the driver in pounds; h < 79 and w < 295.�

� E�� Sample answer: 72 in. and 220 lb � F�� Yes. (50, 180) is a solution because the point falls in the shaded region. �

Solve each system of inequalities by graphing.����y < 6

y > x + 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y �����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > x�����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 6 and y > x + 3. OverlayWKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����y ��� y ��x í 5

62/87,21���Graph each inequality. The graph of y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�y ���x í 5 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y �����DQG�y ���x í 5. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y > 6x + 2

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ���x������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y > 6x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y > 6x�������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����y < 5x í 2 y > í6x + 2

62/87,21���Graph each inequality. The graph of y < 5x í 2 is dashed and is not included in the graph of the solution. The graph of y > í6x + 2 is also dashed and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 2 and y > í6x + 2. The solution region is the darkest shaded area in the graph below. �

����2x í y ���� x í y �� í1

62/87,21���Graph each inequality. The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��7KH�JUDSK�RI�x í y� í1is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x í y �����DQG�x í y �� í1. The solution region is the darkest shaded area in the graph below.

����3x í y > í5 5x í y < 9

62/87,21���Graph each inequality. The graph of 3x í y > í5 is dashed and is not included in the graph of the solution. The graph of 5x í y < 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y > í5 and 5x í y < 9. The solution region is the darkest shaded area in the graph below.

����y ��x + 10 y ��x í 3

62/87,21���Graph each inequality. The graph of y ���x + 10 is solid. The graph of y ���x í 3 is also solid. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ���x + 10 and y ���x í 3. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y < 5x í 5 y > 5x + 9

62/87,21���Graph each inequality. The graph of y < 5x í 5 is dashed and is not included in the graph of the solution. The graph of y > 5x + 9 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 5x í 5 and y > 5x + 9. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����y ���x í 5 3x í y > í4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x í y > í��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x í 5 and 3x í y > í4. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����4x + y > í1 y < í4x + 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x + y > �LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < í4x + 1 is also dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y > í1 and y < í4x + 1.2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� The solution region is shaded in the graph below.

����3x í y �� í2 y < 3x + 4

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x í y �� í��LV�VROLG��

The graph of y < 3x�����LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

Because the lines are parallel and the solution of the system must satisfy 3x í y �� í2 and y < 3x + 4, the line 3x í y � í2 is included in the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x í y �� í2 and y < 3x������ 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ�� �

� The solution region is shaded in the graph below.

����y > 2x í 3 2x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y > 2x í ��LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 2x í y �����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y > 2x í 3 and 2x í y ������2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

� � The solution region is shaded in the graph below.

����5x � y < �6 3x � y ��� �

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 5x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 5x � y < �6 and 3x � y �����

This region is shaded in the graph below.

����x � y ��� y < 3x

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x � y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of y < 3x�LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x � y ����DQG�y < 3x��

This region is shaded in the graph below.

����4x + y < í2 y > í4x

62/87,21���Graph each inequality. The graph of 4x + y < í2 is dashed and is not included in the graph of the solution. The graph of y > í4x is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x + y < í2 and y > í4x. This region is shaded in the graph below. Notice that the graphs do not intersect, so there is no solution.

����,&(�5,1.6���,FH�UHVXUIDFHUV�DUH�XVHG�IRU�ULQNV�RI�DW�OHDVW������VTXDUH�IHHW�DQG�XS�WR��������VTXDUH�IHHW��7KH�price ranges from as little as $10,000 to as much as $150,000. � D�� Define the variables, and write a system of inequalities to represent this situation. Then graph the system. � E�� Name one possible solution. � F�� Is (15,000, 30,000) a solution? Explain.

62/87,21���a. Let x represent square footage of the rink. Let y represent the price of the resurfacing. ;

. �

� E�� Sample answer: an ice resurfacer for a rink of 5000 ft2 and a price of $20,000 � F�� Yes. The point (15,000, 30,000) satisfies each inequality. �

����&&66�02'(/,1*� Josefina works between 10 and 30 hours per week at a pizzeria. She earns $6.50 an hour, butcan earn tips when she delivers pizzas. � D�� Write a system of inequalities to represent the dollars d she could earn for working h hours in a week. � E�� Graph this system. � F�� If Josefina received $17.50 in tips and earned a total of $180 for the week, how many hours did she work?

62/87,21���D�� d ������h �����h����� � b.

� F�� Because Josefina received $17.50 in tips, her hourly wages can be found by $180 ± $17.50 = $162.50. Next, GLYLGH���������E\�������SHU�KRXU�WR�GHWHUPLQH�KRZ�PDQ\�KRXUV�VKH�ZRUNHG����������·������� �����VR�-RVHILQD�worked for 25 hours.

Solve each system of inequalities by graphing.����x + y ���

x + y ���

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x + y ����LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of x + y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x + y ����DQG�x + y ����

�7KLV�UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ�

����3x � y < �2 3x � y < 1

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 3x � y < ���LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 3x � y < �2 and 3x � y < 1.

This region is shaded in the graph below.

����2x � y ���11 3x � y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2x � y ������LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 3x � y ����LV�DOVR�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2x � y ���11 and 3x � y ������

This region is shaded in the graph below.

����y < 4x + 13 4x í y ����

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y < 4x������LV�GDVKHG�DQG�LV�QRW�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The graph of 4x í y �����LV�VROLG���

Because the lines are parallel and the solution of the system must satisfy y < 4x + 13 and 4x í y ������WKH�OLQH��x í y ����LV�LQFOXGHG�LQ�WKH�VROXWLRQ��2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y < 4x + 13 and 4x í y ������ � The solution region is shaded in the graph below.

����4x í y < í3 y ���x í 6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 4x í y < í3 is dashed and is not included in the graph of the solution.

� The graph of y ����x í 6 is solid and is included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of 4x í y < í3 and y ����x í 6. 2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

7KH�VROXWLRQ��UHJLRQ�LV�VKDGHG�LQ�WKH�JUDSK�EHORZ��%HFDXVH�WKH�OLQHV�DUH�SDUDOOHO�DQG�WKH�VROXWLRQ�RI�WKH�V\VWHP�PXVW�satisfy 4x ± y < í3 and y ����x ± 6, the line y ����x ± 6 is not included in the solution.

����y ���x + 7 y < 2x í 3

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of y ����x + 7 is solid and is included in the graph of the solution��

The graph of y < 2x í 3 is dashed and is not included in the graph of the solution��

The solution of the system is the set of ordered pairs in the intersection of the graphs of y ����x + 7 and y < 2x í ���2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below. Because the lines are parallel and the solution of the system must satisfy y ����x + 7 and y < 2x í 3, the line y ����x + 7 is not included in the solution.

����

62/87,21���Graph each inequality. The graph of is dashed and is not included in the graph of the solution. The graph of is solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of and y ����x + 2. The solution region is the darkest shaded area in the graph below.

����2y ���x x í 3y > í6

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of 2y ���x is solid and is included in the graph of the solution.�

The graph of x í 3y > í6 is dashed and is not included in the graph of the solution.

The solution of the system is the set of ordered pairs in the intersection of the graphs of 2y ���x and x í 3y > í6. �2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����x í 5y > í15 5y ���x í 5

62/87,21���*UDSK�HDFK�LQHTXDOLW\�� The graph of x í 5y > í15 is dashed and is not included in the graph of the solution.

� The graph of 5y ���x í ��LV�VROLG�DQG�LV�LQFOXGHG�LQ�WKH�JUDSK�RI�WKH�VROXWLRQ��

The solution of the system is the set of ordered pairs in the intersection of the graphs of x í 5y > í15 and 5y ���x í 5.�2YHUOD\�WKH�JUDSKV�DQG�ORFDWH�WKH�JUHHQ�UHJLRQ��7KLV�LV�WKH�LQWHUVHFWLRQ��

The solution region is shaded in the graph below.

����&/$66�352-(&7� An economics class formed a company to sell school supplies. They would like to sell at least 20 notebooks and 50 pens per week, with a goal of earning at least $60 per week. �

D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph the system. � F�� Name one possible solution.

62/87,21���D�� Let n = the number of notebooks and p = the number of pens; n�������p ������������n + $1.25p ������� b.

� F�� Sample answer: 25 notebooks and 100 pens �

����),1$1&,$/�/,7(5$&<� Opal makes $15 per hour working for a photographer. She also coaches a competitive soccer team for $10 per hour. Opal needs to earn at least $90 per week, but she does not want to work more than 20hours per week. � D�� Define the variables, and write a system of inequalities to represent this situation. � E�� Graph this system. � F�� Give two possible solutions to describe how Opal can meet her goals. � G�� Is (2, 2) a solution? Explain.

62/87,21���D�� Let x = the hours worked for the photographer, let y = the hours coaching, x + y � 20, 15x + 10y � 90. � b.

� F�� Sample answer: 6 hours at the photographer¶s, 10 hours of coaching; 8 hours at the photographer¶s, 10 hours of coaching � G�� No. The point (2, 2) does not fall in the shaded region. She would not earn enough money.

����&+$//(1*(� Create a system of inequalities equivalent to |x_�����

62/87,21���You learned in Lesson 5-5 that the inequality |x_�����PHDQV�WKDW�WKH�GLVWDQFH�EHWZHHQ�x and 0 is less than or equal to 4. On a number line this would be the numbers between -4 and 4 inclusive, as shown below.

Using a pair of linear equations, this would be shown as all the points between the lines x = -4 and x = 4. This could be expressed by the system of equations x � 4 and x � í4.

����5($621,1*� State whether the following statement is sometimes, always, or never true. Explain your answer with an example or counterexample.

Systems of inequalities with parallel boundaries have no solutions.

62/87,21���The following system of equations have parallel boundaries and have no solution since the shaded regions do not overlap. 2x ± y > 3 and 2x ± y < ±1

However, the system given by y �����DQG�y ���±2 also have parallel boundaries but the solution will contain an infinite QXPEHU�RI�SRLQWV�DV�RQH�RI�WKH�VKDGHG�UHJLRQV�FRPSOHWHO\�RYHUODSV�WKH�RWKHU��

The solution would be all the points such that y ���±2. Therefore, it is sometimes true that systems with parallel boundaries have no solution. It is also possible that the solution could be one of the regions determined by the parallel boundaries.

����5($621,1*� Describe the graph of the solution of this system without graphing. 6x í 3y �� í5 6x í 3y �� í5

62/87,21���Both inequalities have the same boundary line, which is included in the solution; however, they are shaded in differentdirections. So, the solution that satisfies both inequalities is only the line.

����23(1�(1'('� One inequality in a system is 3x í y > 4. Write a second inequality so that the system will have no solution.

62/87,21���In order for the system to have no solution, the solution cannot overlap at all. Therefore, an inequality that would result in a system with no solution is 3x í y < í4.

����&&66�35(&,6,21� Graph the system of inequalities. Estimate the area of the solution. �

y ��� y ��x + 4

y � íx + 4

62/87,21���

� To estimate the area, count up the total number of squares and half±squares. There are 6 whole squares and 6 half squares. 6 + 6(0.5) = 9. So, the area is .

����Jacui is beginning an exercise program that involves an intense cardiovascular workout. Her trainer recommends thatfor a person her age, her heart rate should stay within the following range as she exercises. ��It should be higher than 102 beats per minute. ��It should not exceed 174 beats per minute.

� :5,7,1*�,1�0$7+� Explain what each colored region of the graph represents. Explain how shading in various colors can help to clearly show the solution set of a system of inequalities.

62/87,21���Sample answer: The yellow region represents the beats per minute below the target heart rate. The blue region represents the beats per minute above the target heart rate. The green region represents the beats per minute within the target heart rate. Shading in different colors clearly shows the overlapping solution set of the system of inequalities.

����(;7(1'('�5(63216(� To apply for a scholarship, you must have a minimum of 20 hours of community service and a grade±point average of at least 3.75. Another scholarship requires at least 40 hours of community service and a minimum grade±point average of 3.0. � D�� Write a system of inequalities to represent the credentials you must have to apply for both scholarships. � E�� Graph the system of inequalities. � F�� If you are eligible for both scholarships, give one possible solution.

62/87,21���D�� c �����DQG�g������� c �����DQG�g������ � b.

� F�� Sample answer: 40 community service hours and a 3.75 grade points average

����*(20(75<� What is the measure of Ә1? �

� $� ��� � %� ��� � &� ��� � '� ���

62/87,21���The three angles add up 180o because it is a straight line. Let m = the measure of .�

� So, the correct choice is D.

����*(20(75<� What is the volume of the triangular prism?�

� )� 120 cm3 � *� 96 cm3 � +� 48 cm3 � -� 30 cm3

62/87,21���To find the volume of the triangular prism use the formula:�

� So, the correct choice is H.

����Ten pounds of fresh tomatoes make about 15 cups of cooked tomatoes. How many cups of cooked tomatoes does one pound of fresh tomatoes make? �

$� 1 cups

� %� 3 cups � &� 4 cups � '� 5 cups

62/87,21���Let x represent the number of cups of cooked tomatoes. �

One pound of fresh tomatoes will make 1 FXSV�RI�FRRNHG�WRPDWRHV��

� So, the correct choice is A.

����&+(0,675<� Orion Labs needs to make 500 gallons of 34% acid solution. The only solutions available are a 25% acid solution and a 50% acid solution. Write and solve a system of equations to find the number of gallons of each solution that should be mixed to make the 34% solution.

62/87,21���The total amount of acid is 34% of 500 = 170. The acid is 25% of one solution (x) and 50% of another (y). So, one equation is 0.25x + 0.5y = 170. � The total number of gallons is 500, which is the sum of the solutions. Therefore, the second equation is x + y = 500. x + y = 500, 0.25x + 0.5y = 170 � Solve by substitution. First solve equation 1 for x��

� 6XEVWLWXWH�LQWR�HTXDWLRQ����

� Substitute 180 for y in either equation to find the value of x.

� 320 gal of the 25% solution and 180 gal of the 50% solution should be mixed to make the 34% solution.

Use elimination to solve each system of equations.����x + y = 7

2x + y = 11

62/87,21���Multiply equation 2 by ±1, then solve by elimination using addition.�

� Substitute in either equation and solve for y �� �

� The solution is (4, 3).

����a ± b = 9 7a + b = 7

62/87,21���

� Substitute into either equation and solve for b���

� The solution is (2, ±7).

����q + 4r = í8 3q + 2r = 6

62/87,21���First, multiply the second equation by ±2.

� Substitute into either equation and solve for r�� �

� The solution is (4, ±3).

����(17(57$,10(17� A group of 11 adults and children bought tickets for the baseball game. If the total cost was $156, how many of each type of ticket did they buy? $156, how many of each type of ticket did they buy?

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Notice that if you multiply the first equation by 12, the y±terms are the same, so subtract the equations.�

� Now, substitute 8 in for x into either equation and solve for y . �

� So, 8 adult tickets and 3 children¶s tickets were purchased.

Graph each inequality.����4x í �����y

62/87,21���Solve for y in terms of x.

%HFDXVH�WKH�LQHTXDOLW\�LQYROYHV����JUDSK�WKH�ERXQGDU\�XVLQJ�D�VROLG�OLQH��&KRRVH��������DV�D�WHVW�SRLQW�

Since í1 is not greater than or equal to 0, shade the half±plane that does not contain (0, 0).

����9x í 3y < 0

62/87,21���Solve for y in terms of x.

Because the inequality involves <, graph the boundary using a dashed line. Choose (1, 1) as a test point

Since 3 is not less than 01 shade the half±plane that does not contain (1, 1).

����2y �� í4x í 6

62/87,21���Solve for y in terms of x.

%HFDXVH�WKH�LQHTXDOLW\�LQYROYHV����JUDSK�WKH�ERXQGDU\�XVLQJ�D�VROLG�OLQH��&KRRVH��������DV�D�WHVW�SRLQW�

Since 0 is not less than or equal to í3, shade the half±plane that does not contain (0, 0).

Evaluate each expression.����33

62/87,21���

����24

62/87,21���

����(í4)3

62/87,21���

eSolutions Manual - Powered by Cognero Page 45

6-6 Systems of Inequalities

$156, how many of each type of ticket did they buy?

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Notice that if you multiply the first equation by 12, the y±terms are the same, so subtract the equations.�

� Now, substitute 8 in for x into either equation and solve for y . �

� So, 8 adult tickets and 3 children¶s tickets were purchased.

Graph each inequality.����4x í �����y

62/87,21���Solve for y in terms of x.

%HFDXVH�WKH�LQHTXDOLW\�LQYROYHV����JUDSK�WKH�ERXQGDU\�XVLQJ�D�VROLG�OLQH��&KRRVH��������DV�D�WHVW�SRLQW�

Since í1 is not greater than or equal to 0, shade the half±plane that does not contain (0, 0).

����9x í 3y < 0

62/87,21���Solve for y in terms of x.

Because the inequality involves <, graph the boundary using a dashed line. Choose (1, 1) as a test point

Since 3 is not less than 01 shade the half±plane that does not contain (1, 1).

����2y �� í4x í 6

62/87,21���Solve for y in terms of x.

%HFDXVH�WKH�LQHTXDOLW\�LQYROYHV����JUDSK�WKH�ERXQGDU\�XVLQJ�D�VROLG�OLQH��&KRRVH��������DV�D�WHVW�SRLQW�

Since 0 is not less than or equal to í3, shade the half±plane that does not contain (0, 0).

Evaluate each expression.����33

62/87,21���

����24

62/87,21���

����(í4)3

62/87,21���

eSolutions Manual - Powered by Cognero Page 46

6-6 Systems of Inequalities

$156, how many of each type of ticket did they buy?

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Notice that if you multiply the first equation by 12, the y±terms are the same, so subtract the equations.�

� Now, substitute 8 in for x into either equation and solve for y . �

� So, 8 adult tickets and 3 children¶s tickets were purchased.

Graph each inequality.����4x í �����y

62/87,21���Solve for y in terms of x.

%HFDXVH�WKH�LQHTXDOLW\�LQYROYHV����JUDSK�WKH�ERXQGDU\�XVLQJ�D�VROLG�OLQH��&KRRVH��������DV�D�WHVW�SRLQW�

Since í1 is not greater than or equal to 0, shade the half±plane that does not contain (0, 0).

����9x í 3y < 0

62/87,21���Solve for y in terms of x.

Because the inequality involves <, graph the boundary using a dashed line. Choose (1, 1) as a test point

Since 3 is not less than 01 shade the half±plane that does not contain (1, 1).

����2y �� í4x í 6

62/87,21���Solve for y in terms of x.

%HFDXVH�WKH�LQHTXDOLW\�LQYROYHV����JUDSK�WKH�ERXQGDU\�XVLQJ�D�VROLG�OLQH��&KRRVH��������DV�D�WHVW�SRLQW�

Since 0 is not less than or equal to í3, shade the half±plane that does not contain (0, 0).

Evaluate each expression.����33

62/87,21���

����24

62/87,21���

����(í4)3

62/87,21���

eSolutions Manual - Powered by Cognero Page 47

6-6 Systems of Inequalities

$156, how many of each type of ticket did they buy?

62/87,21���Let x represent the number of adult tickets and y represent the number of children¶s tickets. �

� Notice that if you multiply the first equation by 12, the y±terms are the same, so subtract the equations.�

� Now, substitute 8 in for x into either equation and solve for y . �

� So, 8 adult tickets and 3 children¶s tickets were purchased.

Graph each inequality.����4x í �����y

62/87,21���Solve for y in terms of x.

%HFDXVH�WKH�LQHTXDOLW\�LQYROYHV����JUDSK�WKH�ERXQGDU\�XVLQJ�D�VROLG�OLQH��&KRRVH��������DV�D�WHVW�SRLQW�

Since í1 is not greater than or equal to 0, shade the half±plane that does not contain (0, 0).

����9x í 3y < 0

62/87,21���Solve for y in terms of x.

Because the inequality involves <, graph the boundary using a dashed line. Choose (1, 1) as a test point

Since 3 is not less than 01 shade the half±plane that does not contain (1, 1).

����2y �� í4x í 6

62/87,21���Solve for y in terms of x.

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Since 0 is not less than or equal to í3, shade the half±plane that does not contain (0, 0).

Evaluate each expression.����33

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6-6 Systems of Inequalities