2 6 inequalities
TRANSCRIPT
![Page 1: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/1.jpg)
Inequalities
![Page 2: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/2.jpg)
We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left.
Inequalities
![Page 3: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/3.jpg)
We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
Inequalities
![Page 4: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/4.jpg)
We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
2/3
Inequalities
![Page 5: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/5.jpg)
We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
2/3 2½
Inequalities
![Page 6: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/6.jpg)
We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
2/3 2½ π 3.14..
Inequalities
–π –3.14..
![Page 7: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/7.jpg)
We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
2/3 2½ π 3.14..
This line with each position addressed by a real number is called the real (number) line.
Inequalities
–π –3.14..
![Page 8: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/8.jpg)
We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
2/3 2½ π 3.14..
This line with each position addressed by a real number is called the real (number) line.
Inequalities
–π –3.14..
Given two numbers corresponding to two points on the real line, we define the number to the right to be greater than the number to the left.
![Page 9: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/9.jpg)
We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
2/3 2½ π 3.14..
This line with each position addressed by a real number is called the real (number) line.
Inequalities
+–RL
–π –3.14..
Given two numbers corresponding to two points on the real line, we define the number to the right to be greater than the number to the left.
![Page 10: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/10.jpg)
We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
2/3 2½ π 3.14..
This line with each position addressed by a real number is called the real (number) line.
Inequalities
+–R
We write this as L < R and called this the natural form because it corresponds to their respective positions on the real line.
L <
–π –3.14..
Given two numbers corresponding to two points on the real line, we define the number to the right to be greater than the number to the left.
![Page 11: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/11.jpg)
We associate each real number with a position on a line, positive numbers to the right and negative numbers to the left.
-2 20 1 3+
-1-3–
2/3 2½ π 3.14..
This line with each position addressed by a real number is called the real (number) line.
Inequalities
+–R
We write this as L < R and called this the natural form because it corresponds to their respective positions on the real line. This relation may also be written as R > L (less preferable).
L <
–π –3.14..
Given two numbers corresponding to two points on the real line, we define the number to the right to be greater than the number to the left.
![Page 12: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/12.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statementsInequalities
![Page 13: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/13.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statementsand –2 < –5 , 5 < 3 are false statements.
Inequalities
![Page 14: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/14.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statementsand –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x".
![Page 15: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/15.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statementsand –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a).
![Page 16: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/16.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statementsand –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a). In picture,
+–a
open dot
a < x
![Page 17: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/17.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statementsand –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a). In picture,
+–a
open dot
If we want all the numbers x greater than or equal to a (including a), we write it as a < x.
a < x
![Page 18: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/18.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statementsand –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a). In picture,
+–a
open dot
If we want all the numbers x greater than or equal to a (including a), we write it as a < x. In picture
+–a
solid dot
a < x
a < x
![Page 19: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/19.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statementsand –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a). In picture,
+–a
open dot
If we want all the numbers x greater than or equal to a (including a), we write it as a < x. In picture
+–a
solid dot
a < x
a < x
The numbers x fit the description a < x < b where a < b are all the numbers x between a and b.
![Page 20: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/20.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statementsand –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a). In picture,
+–a
open dot
If we want all the numbers x greater than or equal to a (including a), we write it as a < x. In picture
+–a
solid dot
a < x
a < x
The numbers x fit the description a < x < b where a < b are all the numbers x between a and b.
+–a a < x < b b
![Page 21: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/21.jpg)
Example A. 2 < 4, –3< –2, 0 > –1 are true statementsand –2 < –5 , 5 < 3 are false statements.
Inequalities
If we want all the numbers greater than 5, we may denote them as "all number x where 5 < x". In general, we write "a < x" for all the numbers x greater than a (excluding a). In picture,
+–a
open dot
If we want all the numbers x greater than or equal to a (including a), we write it as a < x. In picture
+–a
solid dot
a < x
a < x
The numbers x fit the description a < x < b where a < b are all the numbers x between a and b. A line segment as such is called an interval.
+–a a < x < b b
![Page 22: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/22.jpg)
Example B.a. Draw –1 < x < 3.
Inequalities
![Page 23: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/23.jpg)
Example B.a. Draw –1 < x < 3.
Inequalities
It’s in the natural form.
![Page 24: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/24.jpg)
Example B.a. Draw –1 < x < 3.
Inequalities
It’s in the natural form. Mark the numbers and x on the linein order accordingly.
![Page 25: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/25.jpg)
Example B.a. Draw –1 < x < 3.
0 3+
-1– x
Inequalities
It’s in the natural form. Mark the numbers and x on the linein order accordingly.
![Page 26: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/26.jpg)
Example B.a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
Inequalities
It’s in the natural form. Mark the numbers and x on the linein order accordingly.
![Page 27: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/27.jpg)
Example B.a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
Inequalities
It’s in the natural form. Mark the numbers and x on the linein order accordingly.
Put it in the natural form –3 < x < 0.
![Page 28: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/28.jpg)
Example B.a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
Inequalities
It’s in the natural form. Mark the numbers and x on the linein order accordingly.
Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly.
![Page 29: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/29.jpg)
Example B.a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
0+
-3–
x
Inequalities
It’s in the natural form. Mark the numbers and x on the linein order accordingly.
Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly.
![Page 30: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/30.jpg)
Example B.a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
0+
-3–
x
Expressions such as 2 < x > 3 or 2 < x < –3 do not have any solution.
Inequalities
It’s in the natural form. Mark the numbers and x on the linein order accordingly.
Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly.
![Page 31: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/31.jpg)
Example B.a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
0+
-3–
x
Expressions such as 2 < x > 3 or 2 < x < –3 do not have any solution.
Inequalities
Adding or subtracting the same quantity to both retains the inequality sign,
It’s in the natural form. Mark the numbers and x on the linein order accordingly.
Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly.
![Page 32: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/32.jpg)
Example B.a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
0+
-3–
x
Expressions such as 2 < x > 3 or 2 < x < –3 do not have any solution.
Inequalities
Adding or subtracting the same quantity to both retains the inequality sign, i.e. if a < b, then a ± c < b ± c.
It’s in the natural form. Mark the numbers and x on the linein order accordingly.
Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly.
![Page 33: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/33.jpg)
Example B.a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
0+
-3–
x
Expressions such as 2 < x > 3 or 2 < x < –3 do not have any solution.
Inequalities
For example 6 < 12, then 6 + 3 < 12 + 3.
Adding or subtracting the same quantity to both retains the inequality sign, i.e. if a < b, then a ± c < b ± c.
It’s in the natural form. Mark the numbers and x on the linein order accordingly.
Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly.
![Page 34: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/34.jpg)
Example B.a. Draw –1 < x < 3.
0 3+
-1– x
b. Draw 0 > x > –3
0+
-3–
x
Expressions such as 2 < x > 3 or 2 < x < –3 do not have any solution.
Inequalities
For example 6 < 12, then 6 + 3 < 12 + 3. We use the this fact to solve inequalities.
Adding or subtracting the same quantity to both retains the inequality sign, i.e. if a < b, then a ± c < b ± c.
It’s in the natural form. Mark the numbers and x on the linein order accordingly.
Put it in the natural form –3 < x < 0. Then mark the numbers and x in order accordingly.
![Page 35: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/35.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
Inequalities
![Page 36: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/36.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3
Inequalities
![Page 37: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/37.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
Inequalities
![Page 38: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/38.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
![Page 39: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/39.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c.
![Page 40: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/40.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign,
![Page 41: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/41.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b
![Page 42: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/42.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc.
![Page 43: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/43.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc. For example 6 < 12 is true,
![Page 44: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/44.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc. For example 6 < 12 is true, then multiplying by 3 3*6 < 3*12
![Page 45: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/45.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc. For example 6 < 12 is true, then multiplying by 3 3*6 < 3*12 or 18 < 36 is also true.
![Page 46: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/46.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc.
Example D. Solve 3x > 12 and draw the solution.
For example 6 < 12 is true, then multiplying by 3 3*6 < 3*12 or 18 < 36 is also true.
![Page 47: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/47.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc.
Example D. Solve 3x > 12 and draw the solution.
3x > 12
For example 6 < 12 is true, then multiplying by 3 3*6 < 3*12 or 18 < 36 is also true.
![Page 48: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/48.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc.
Example D. Solve 3x > 12 and draw the solution.
3x > 12 divide by 3 and keep the inequality sign 3x/3 > 12/3
For example 6 < 12 is true, then multiplying by 3 3*6 < 3*12 or 18 < 36 is also true.
![Page 49: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/49.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc.
Example D. Solve 3x > 12 and draw the solution.
3x > 12 divide by 3 and keep the inequality sign 3x/3 > 12/3 x > 4 or 4 < x
For example 6 < 12 is true, then multiplying by 3 3*6 < 3*12 or 18 < 36 is also true.
![Page 50: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/50.jpg)
Example C. Solve x – 3 < 12 and draw the solution.
x – 3 < 12 add 3 to both sides x – 3 + 3 < 12 + 3 x < 15
0 15+–
Inequalities
x
A number c is positive means that 0 < c. We may multiply or divide a positive number to the inequality and keep the same inequality sign, i.e. if 0 < c and a < b, then ac < bc.
Example D. Solve 3x > 12 and draw the solution.
3x > 12 divide by 3 and keep the inequality sign 3x/3 > 12/3 x > 4 or 4 < x
40+–
For example 6 < 12 is true, then multiplying by 3 3*6 < 3*12 or 18 < 36 is also true.
x
![Page 51: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/51.jpg)
A number c is negative means c < 0.
Inequalities
![Page 52: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/52.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign,
Inequalities
![Page 53: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/53.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Inequalities
![Page 54: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/54.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Inequalities
For example 6 < 12 is true.
![Page 55: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/55.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Inequalities
For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true.
![Page 56: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/56.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Inequalities
For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
![Page 57: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/57.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Inequalities
60+–
12<
For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
![Page 58: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/58.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Inequalities
60+–
12–6 <
For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
![Page 59: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/59.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Inequalities
60+–
12–6–12 <<
For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
![Page 60: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/60.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Example E. Solve –x + 2 < 5 and draw the solution.
Inequalities
60+–
12–6–12 <<
For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
![Page 61: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/61.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Example E. Solve –x + 2 < 5 and draw the solution.
–x + 2 < 5
Inequalities
60+–
12–6–12 <<
For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
![Page 62: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/62.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Example E. Solve –x + 2 < 5 and draw the solution.
–x + 2 < 5 subtract 2 from both sides –x < 3
Inequalities
60+–
12–6–12 <<
For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
![Page 63: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/63.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Example E. Solve –x + 2 < 5 and draw the solution.
–x + 2 < 5 subtract 2 from both sides –x < 3 multiply by –1 to get x, reverse the inequality –(–x) > –3 x > –3
Inequalities
60+–
12–6–12 <<
For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
![Page 64: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/64.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Example E. Solve –x + 2 < 5 and draw the solution.
–x + 2 < 5 subtract 2 from both sides –x < 3 multiply by –1 to get x, reverse the inequality –(–x) > –3 x > –3 or –3 < x
Inequalities
60+–
12–6–12 <<
For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
![Page 65: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/65.jpg)
A number c is negative means c < 0. Multiplying or dividing by an negative number reverses the inequality sign, i.e. if c < 0 and a < b then ca > cb .
Example E. Solve –x + 2 < 5 and draw the solution.
–x + 2 < 5 subtract 2 from both sides –x < 3 multiply by –1 to get x, reverse the inequality –(–x) > –3 x > –3 or –3 < x
0+
-3–
Inequalities
60+–
12–6–12 <<
For example 6 < 12 is true. If we multiply –1 to both sides then (–1)6 > (–1)12 – 6 > –12 which is true. Multiplying by –1 switches the left-right positions of the originals.
![Page 66: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/66.jpg)
To solve inequalities:1. Simplify both sides of the inequalities
Inequalities
![Page 67: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/67.jpg)
To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides
Inequalities
![Page 68: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/68.jpg)
To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule).
Inequalities
![Page 69: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/69.jpg)
To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x.
Inequalities
![Page 70: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/70.jpg)
To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around.
Inequalities
![Page 71: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/71.jpg)
To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. This rule can be avoided by keeping the x-term positive.
Inequalities
![Page 72: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/72.jpg)
To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. This rule can be avoided by keeping the x-term positive.
Inequalities
Example F. Solve 3x + 5 > x + 9
![Page 73: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/73.jpg)
To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. This rule can be avoided by keeping the x-term positive.
Inequalities
Example F. Solve 3x + 5 > x + 9
3x + 5 > x + 9 move the x and 5, change side-change sign
![Page 74: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/74.jpg)
To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. This rule can be avoided by keeping the x-term positive.
Inequalities
Example F. Solve 3x + 5 > x + 9
3x + 5 > x + 9 move the x and 5, change side-change sign3x – x > 9 – 5
![Page 75: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/75.jpg)
To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. This rule can be avoided by keeping the x-term positive.
Inequalities
Example F. Solve 3x + 5 > x + 9
3x + 5 > x + 9 move the x and 5, change side-change sign3x – x > 9 – 5 2x > 4
![Page 76: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/76.jpg)
To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. This rule can be avoided by keeping the x-term positive.
Inequalities
Example F. Solve 3x + 5 > x + 9
3x + 5 > x + 9 move the x and 5, change side-change sign3x – x > 9 – 5 2x > 4 div. 2
2x 2
4 2>
![Page 77: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/77.jpg)
To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. This rule can be avoided by keeping the x-term positive.
Inequalities
Example F. Solve 3x + 5 > x + 9
3x + 5 > x + 9 move the x and 5, change side-change sign3x – x > 9 – 5 2x > 4 div. 2
2x 2
4 2>
x > 2 or 2 < x
![Page 78: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/78.jpg)
To solve inequalities:1. Simplify both sides of the inequalities2. Gather the x-terms to one side and the number-terms to the other sides (use the “change side-change sign” rule). 3. Multiply or divide to get x. If we multiply or divide by negative numbers to both sides, the inequality sign must be turned around. This rule can be avoided by keeping the x-term positive.
Inequalities
Example F. Solve 3x + 5 > x + 9
3x + 5 > x + 9 move the x and 5, change side-change sign3x – x > 9 – 5 2x > 4 div. 2
20+–
2x 2
4 2>
x > 2 or 2 < x
![Page 79: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/79.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
Inequalities
![Page 80: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/80.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side
Inequalities
![Page 81: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/81.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x
Inequalities
![Page 82: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/82.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18
Inequalities
![Page 83: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/83.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x
Inequalities
![Page 84: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/84.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x – 12 > 3x
Inequalities
![Page 85: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/85.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x – 12 > 3x–12
33x 3
>
div. by 3 (no need to switch >)
Inequalities
![Page 86: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/86.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x – 12 > 3x–12
33x 3
>
–4 > x or x < –4
div. by 3 (no need to switch >)
Inequalities
![Page 87: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/87.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x – 12 > 3x
0+
–12 3
3x 3
>
-4
div. by 3 (no need to switch >)
Inequalities
–4 > x or x < –4
![Page 88: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/88.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x – 12 > 3x
0+
–12 3
3x 3
>
-4
div. by 3 (no need to switch >)
Inequalities
We also have inequalities in the form of intervals.
–4 > x or x < –4
![Page 89: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/89.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x – 12 > 3x
0+
–12 3
3x 3
>
-4
div. by 3 (no need to switch >)
Inequalities
We also have inequalities in the form of intervals. We solve them by +, –, * , / to all three parts of the inequalities.
–4 > x or x < –4
![Page 90: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/90.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x – 12 > 3x
0+
–12 3
3x 3
>
-4
div. by 3 (no need to switch >)
Inequalities
We also have inequalities in the form of intervals. We solve them by +, –, * , / to all three parts of the inequalities. Again, we + or – remove the number term in the middle first,
–4 > x or x < –4
![Page 91: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/91.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x – 12 > 3x
0+
–12 3
3x 3
>
-4
div. by 3 (no need to switch >)
Inequalities
We also have inequalities in the form of intervals. We solve them by +, –, * , / to all three parts of the inequalities. Again, we + or – remove the number term in the middle first, then divide or multiply to get x.
–4 > x or x < –4
![Page 92: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/92.jpg)
Example G. Solve 3(2 – x) > 2(x + 9) – 2x
3(2 – x) > 2(x + 9) – 2x simplify each side6 – 3x > 2x + 18 – 2x6 – 3x > 18 move 18 and –3x (change sign)6 – 18 > 3x – 12 > 3x
0+
–12 3
3x 3
>
-4
div. by 3 (no need to switch >)
Inequalities
We also have inequalities in the form of intervals. We solve them by +, –, * , / to all three parts of the inequalities. Again, we + or – remove the number term in the middle first, then divide or multiply to get x. The answer is an interval of numbers.
–4 > x or x < –4
![Page 93: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/93.jpg)
Example H. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw
Inequalities
![Page 94: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/94.jpg)
Example H. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw.
12 > –2x + 6 > –4 subtract 6 –6 –6 –6
Inequalities
![Page 95: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/95.jpg)
Example H. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw.
12 > –2x + 6 > –4 subtract 6 –6 –6 –6 6 > –2x > –10
Inequalities
![Page 96: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/96.jpg)
Example H. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw.
12 > –2x + 6 > –4 subtract 6 –6 –6 –6 6 > –2x > –10 div. by -2, switch inequality sign 6 -2
-2x -2<
-10 -2
<
Inequalities
![Page 97: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/97.jpg)
Example H. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw.
12 > –2x + 6 > –4 subtract 6 –6 –6 –6 6 > –2x > –10
-3 < x < 5
div. by -2, switch inequality sign 6 -2
-2x -2<
-10 -2
<
Inequalities
![Page 98: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/98.jpg)
Example H. (Interval Inequality)
Solve 12 > –2x + 6 > –4 and draw.
12 > –2x + 6 > –4 subtract 6 –6 –6 –6 6 > –2x > –10
0+
-3 < x < 5
5
div. by -2, switch inequality sign 6 -2
-2x -2<
-10 -2
<
-3
Inequalities
![Page 99: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/99.jpg)
InequalitiesExercise. A. Draw the following Inequalities. Indicate clearly whether the end points are included or not.1. x < 3 2. –5 ≤ x 3. x < –8 4. x ≤ 12 B. Write in the natural form then draw them.5. x ≥ 3 6. –5 > x 7. x ≥ –8 8. x > 12 C. Draw the following intervals, state so if it is impossible.
9. 6 > x ≥ 3 10. –5 < x ≤ 2 11. 1 > x ≥ –8 12. 4 < x ≤ 2
13. 6 > x ≥ 8
14. –5 > x ≤ 2 15. –7 ≤ x ≤ –3 16. –7 ≤ x ≤ –9D. Solve the following Inequalities and draw the solution.17. x + 5 < 3
18. –5 ≤ 2x + 3 19. 3x + 1 < –8 20. 2x + 3 ≤ 12 – x 21. –3x + 5 ≥ 1 – 4x
22. 2(x + 2) ≥ 5 – (x – 1) 23. 3(x – 1) + 2 ≤ – 2x – 9
24. –2(x – 3) > 2(–x – 1) + 3x 25. –(x + 4) – 2 ≤ 4(x – 1)
26. x + 2(x – 3) < 2(x – 1) – 2
27. –2(x – 3) + 3 ≥ 2(x – 1) + 3x + 13
![Page 100: 2 6 inequalities](https://reader035.vdocuments.mx/reader035/viewer/2022062706/557e154dd8b42a08748b483d/html5/thumbnails/100.jpg)
Inequalities
F. Solve the following interval inequalities.
28. –4 ≤ 2x 29. 7 > 3
–x 30. < –4–xE. Clear the denominator first then solve and draw the solution.
5x 2 3
1 23 2 + ≥ x31. x 4
–3 3
–4 – 1 > x32.
x 2 83 3
45 – ≤ 33. x 8 12
–5 7 1 + > 34.
x 2 3–3 2
3 4
41 – + x35. x 4 6
5 53
–1 – 2 + < x36.
x 12 27 3
6 1
43 – – ≥ x37.
≤
40. – 2 < x + 2 < 5 41. –1 ≥ 2x – 3 ≥ –11
42. –5 ≤ 1 – 3x < 10 43. 11 > –1 – 3x > –7
38. –6 ≤ 3x < 12 39. 8 > –2x > –4