graphing linear inequalities in two variables. an example of a linear inequality in two variables is...
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Graphing Linear Inequalities in Two Variables
An example of a linear inequality in two variables is x - 3y ≤ 6. The solution of an inequality in two variables, x and y, is an ordered pair (x, y) that produces a true statement when substituted into the inequality. Which ordered pair is Which ordered pair is NOTNOT a solution of a solution of x - 3y ≤ 6x - 3y ≤ 6??
A. (0,0) B. (6,-1) C. (10, 3) D. (-1,2)A. (0,0) B. (6,-1) C. (10, 3) D. (-1,2)
x - 3y ≤ 6x - 3y ≤ 6
Substitute each point into the inequality. If the statement is true then it is a solution.
(0) – 3(0) ≤ 6(0) – 3(0) ≤ 6True, therefore(0,0) is a solution.
0 ≤ 60 ≤ 6
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Expressions of the type x + 2y ≤ 8 and 3x – y > 6are called linear inequalities in two variables.
A solution of a linear inequality in two variables is an ordered pair (x, y) which makes the inequality true.
Example: (1, 3) is a solution to x + 2y ≤ 8 since (1) + 2(3) ≤ 8 7 ≤ 8. (Yes, this is true.)
The graph of an inequality in two variables is the set of points that represent all solutions of the inequality.
The BOUNDARY LINE of a linear inequality divides the coordinate plane into two HALF-PLANES. Only one half-plane contains the points that represent the solutions to the inequality.
The solution set, or feasible set, of a linear inequality in two variables is the set of all solutions.
The solution set is a half-plane. It consists of the line x + 2y ≤ 8 and all the points below and to its left.
The line is called the boundary line of the half-plane.
Example: The solution set for x + 2y ≤ 8 is the shaded region. x
y
2
2
Some solutions in the solution set are (0,0), (2, -2), and (-4, 2). (6, 4) is not in the solution set!
x
yIf the inequality is ≤ or ≥ , the boundary line is solid; its points are solutions.
Example: The boundary line of the solution set of 3x – y ≥ 2 is solid.
3x – y < 2
3x – y = 2
3x – y > 2
x
y
If the inequality is < or >, the boundary line is dotted; its points are not solutions.
Example: The boundary line of the solution set of x + y < 2 is dotted.
8
x
yExample: For 2x – 3y ≤ 18 graph the boundary line.
A test point can be selected to determine which side of the half-plane to shade. Pick any point that is not on your line.
Use (0, 0) as a test point.
2(0) – 3(0) ≤ 18
0 ≤ 18 Yes/True
Shade towards your test point/Include your test point in the shading!
The solution set/feasible set, is the set of all solutions in the shaded region.
(0, 0)
2-2
To graph the solution set/feasible set for a linear inequality:
Step 2. Select a test point, not on the boundary line, and determine if it is a solution.
Step 3. Shade a half-plane.
Step 1. Graph the boundary line. (Remember to check if your line is a dotted line or a solid line.)
10
x
y
Example: Graph the solution set for x – y > 2.
Step 1.) Graph the boundary line x – y = 2 as a dotted line.
Step 2.) Select a test point not on the line, say (0, 0).
x – y > 2
(0) – (0) > 2
0 > 2 No/False!Step 3.) Since this is a not a solution, shade in the half-plane not containing (0, 0). Or, shade away from your test point.
The solution set/feasible set, is the set of all solutions in the shaded region.
(0, 0)
(2, 0)
(0, -2)
11
Solution sets for inequalities with only one variable can be graphed in the same way. Example: Graph the solution set for x < - 2.
Step 1.) Graph the solution set for x < - 2.
x
y
4
4
- 4
- 4
Step 2.) Select a test point not on the line, say (0, 0).
x < - 20 < - 2 No/False!
Step 3.) Since this is a not a solution, shade in the half-plane not containing (0, 0). Or, shade away from your test point.
The solution set/feasible set, is the set of all solutions in the shaded region.
12
Example: Graph the solution set for x ≥ 4.
x
y
4
4
- 4
- 4
Step 2.) Select a test point not on the line, say (0, 0).
x ≥ 4 (0) ≥ 4 No/False!
Since this is a not a solution, shade in the half-plane not containing (0, 0). Or, shade away from your test point.
The solution set/feasible set, is the set of all solutions in the shaded region.
Step 1.) Graph the solution set for x ≥ 4.
Graph the equation
STEPSTEP 11
Graph the inequality Graph the inequality y > 4x - 3y > 4x - 3..
STEPSTEP 22
Test (0,0) in the original inequality.
STEPSTEP 33Shade the half-plane that contains the point (0,0), because (0,0) is a solution to the inequality.
34 xy 34 xy
3)0(40
True
Graph the equation
STEPSTEP 11
Graph the inequality Graph the inequality x + 2y ≤ 0..
STEPSTEP 22
Test (1,0) in the original inequality.
STEPSTEP 33Shade the half-plane that does not contain the point (1,0), because (1,0) is not a solution to the inequality.
02 yx 02 yx
0)0(21
False
01
Graph the equation
STEPSTEP 11
Graph the inequality Graph the inequality -1 ≤ x + y..
STEPSTEP 22
Test (0,0) in the original inequality.
STEPSTEP 33Shade the half-plane that contains the point (0,0), because (0,0) is a solution to the inequality.
yx 1 yx 1
)0()0(1
True01
Graph an InequalityGraph an Inequality
Graph the equation
STEPSTEP 11
Graph the inequality Graph the inequality x – 3y ≤ 12..
STEPSTEP 22
Test (0,0) in the original inequality.
STEPSTEP 33Shade the half-plane that contains the point (0,0), because (0,0) is a solution to the inequality.
1232 yx 1232 yx
12)0(3)0(2
True120
Graph the equation
STEPSTEP 11
Graph the inequality Graph the inequality y ≥ -3..
STEPSTEP 22
Test (0,0) in the original inequality. Use only the y-coordinate, because the inequality does not have a x-variable.
STEPSTEP 33Shade the half-plane that contains the point (0,0), because (0,0) is a solution to the inequality. 3y
3y
3)0(
True
Graph the equation
STEPSTEP 11
Graph the inequality Graph the inequality xx ≤ -1..
STEPSTEP 22
Test (0,0) in the original inequality. Use only the y-coordinate, because the inequality does not have a x-variable.
STEPSTEP 33Shade the half-plane that does not contain the point (0,0), because (0,0) is not a solution to the inequality.
1x
1x
1)0(
False
Write an inequality for each graph.
a. b.
2 xy
331
xy
Write an inequality for each graph.
c. d.
3x2y
Study Guide 6-6 (In-Class) Skills Practice/Practice Worksheet 6-6
Chapter 6 Test on
d. Suppose your budget for a party allows you to spend no more than $12 on peanuts and cashews. Peanuts cost $2/lb and cashews cost $4/lb. Find three possible combinations of peanuts and cashews you can buy.
x = number of pounds of peanutsy = number of pounds of cashews2x + 4y ≤ 12
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.23
A solution of a system of linear inequalities is an ordered pair that satisfies all the inequalities.
(5, 4) is a solution of x + y > 8.(5, 4) is also a solution of 2x – y ≤ 7.
Since (5, 4) is a solution of both inequalities in the system, it is a solution of the system.
Example: Find a solution for the system .
72
8
yx
yx
The set of all solutions of a system of linear inequalities is called its solution set.
1. Shade the half-plane of solutions for each inequality in the system.
To graph the solution set for a system of linear inequalities in two variables:
2. Shade in the intersection of the half-planes.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.25
x
yGraph the solution set for x + y > 8.
The intersection of these two half-planes is the wedge-shaped region at the top of the diagram.
Graph the solution set for 2x – y ≤ 7.
Example: Graph the solution set for the system
72
8
yx
yx
2
2
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.26
Example: Graph the solution set for the system of
linear inequalities:
Graph the two half-planes.
The two half-planes do not intersect; therefore, the solution set is the empty set.
x
y
2x – 3y ≥ 12
-2x + 3y ≥ 6
632
1232
yx
yx
2
2
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.27
x
y
4
4
- 4
- 4
Example: Graph the solution set for the linear system.
Graph each linear inequality.
The solution set is the intersection of all the half-planes.
1
2
16
332
y
x
yx
yx(1)(1)
(2)
(2)
(3)
(3)
(4)
(4)
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