Graphing Linear Inequalities in Two Variables GOAL: graph a linear inequality in two variables...

Graphing Linear Inequalities in Two

Variables

GOAL: graph a linear inequality in two variables

Chapter 6

Algebra 1

Ms. Mayer

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

2

SOLUTION OF LINEAR INEQUALITIES

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.

Using What We KnowSketch a graph of x + y < 3

Step 1: Put into slope intercept

form

y <-x + 3

Step 2: Graph the line

y = -x + 3

Less than means to the left or below.

To check it, pick any point that is not on the line. (0,0) is an easy point to use.

x + y < 3Substitute 0 for x and y.0 + 0 < 30 < 3 Decide if this is true or false.Is 0 less than 3?If it is true, you shade on the same side of the line of the point you picked. If it is false, you shade on the opposite side of the line where the point you picked lies.

Graphing Linear Inequalities

The graph of a linear inequality in two variables is the graph of all solutions of the inequality.

GRAPHING A LINEAR INEQUALITY

The graph of a linear inequality in two variables is a half-plane. To graph a linear inequality, follow these steps.

1STEP

2STEP

Graph the boundary line of the inequality. Use a dashed line for < or > and a solid line for or .

To decide which side of the boundary line to shade, test a point not on the boundary line to see whether it is a solution of the inequality. Then shade the appropriate half-plane.

The boundary line of the inequality divides the coordinate plane into two half-planes: a shaded region which contains the points that are solutions of the inequality, and an unshaded region which contains the points that are not.

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

6

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.

Graphing a Linear Inequality

Sketch a graph of y 3

Graphing an Inequality in Two Variables

Graph x < 2Step 1: Start by graphing the line x = 2

Now what points would give you less

than 2?

Since it has to be x < 2 we shade everything to

the left of the line.

Graph a) y < –2 and b) x 1 in a coordinate plane.

SOLUTION

Test the point (0, 0). Because (0, 0) is not a solution of the inequality, shade the half-plane below the line.

Graph the boundary line y = –2. Use a dashed line because y < – 2.

Test the point (0, 0). Because (0, 0) is a solution of the inequality, shade the half-plane to the left of the line.

Graph the boundary line x = 1. Use a solid line because x 1.

Some Helpful Hints

•If the sign is > or < the line is dashed

•If the sign is or the line will be solid

When dealing with just x and y.

•If the sign > or the shading either goes up or to the right

•If the sign is < or the shading either goes down or to the left

When dealing with slanted lines

•If it is > or then you shade above

•If it is < or then you shade below the line

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

12

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.

13

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

EXAMPLE 1Which ordered pair is a solution of

5x - 2y ≤ 6?

A. (0, -3)

B. (5, 5)

C. (1, -2)

D. (3, 3)

EXAMPLE 2Graph the inequality x ≤ 4 in a coordinate plane.HINT: Remember HOY VEX.Decide whether to

use a solid or dashed line.

Use (0, 0) as a test point.

Shade where the solutions will be.

y

x

5

5

-5

-5

EXAMPLE 3Graph 3x - 4y > 12 in a coordinate plane.Sketch the boundary line of the graph.

Find the x- and y-intercepts and plot them.

Solid or dashed line?

Use (0, 0) as a test point.

Shade where the solutions are.

y

x

5

5

-5

-5

EXAMPLE 4:USING A NEW TEST POINT

Graph y < 2/5x in a coordinate plane.Sketch the boundary line of the graph.

Find the x- and y-intercept and plot them. Both are the origin!

•Use the line’s slopeto graph another point.

Solid or dashed line?

Use a test pointOTHER than theorigin.

Shade where the solutions are.

y

x

5

5

-5

-5

HOMEWORK