Graphing Linear Inequalities in Two Variables Digital Lesson.

Graphing Linear Inequalities in Two Variables

Digital Lesson

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

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) = 7 ≤ 8.

Solution of Linear Inequalities


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


x

y

x

yIf the inequality is < or >, the boundary line is dotted; its points are not solutions.

If the inequality is ≤ or ≥ , the boundary line is solid; its points are solutions.

Example: The boundary line of the solution set of x + y < 2 is dotted.

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

Example: For 2x – 3y ≤ 18 graph the boundary line.

The solution set is a half-plane.

A test point can be selected to determine which side of the half-plane to shade.

Shade above and to the left of the line.

Use (0, 0) as a test point.

(0, 0) is a solution. So all points on the (0, 0) side of the boundary line are also solutions.

(0, 0)

2-2


To graph the solution set for a linear inequality:

2. Select a test point, not on the boundary line, and determine if it is a solution.

3. Shade a half-plane.

1. Graph the boundary line.


x

y

Example: Graph the solution set for x – y > 2.

1. Graph the boundary line x – y = 2 as a dotted line.

2. Select a test point not on the line, say (0, 0).

(0) – 0 = 0 > 2 is false.

3. Since this is a not a solution, shade in the half-plane not containing (0, 0).

(0, 0)

(2, 0)

(0, -2)


Solution sets for inequalities with only one variable can be graphed in the same way.

Example: Graph the solution set for x < - 2.

x

y

4

4

- 4

- 4

x

y

4

4

- 4

- 4

Example: Graph the solution set for x ≥ 4.


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.


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


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


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)

Graphing Linear Inequalities in Two Variables Digital Lesson.

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