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

3

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)