Intermediate Algebra (11th edition)

SECTION 3.4 Linear Inequalities in Two Variables 175

OBJECTIVE 1 Graph linear inequalities in two variables.In Chapter 2,we

graphed linear inequalities in one variable on the number line. In this section, we

graph linear inequalities in two variables on a rectangular coordinate system.

OBJECTIVES

Linear Inequalities in Two Variables

3.4

1 Graph linear
inequalities in two
variables.

2 Graph the
intersection of two
linear inequalities.

3 Graph the union
of two linear
inequalities.

FIGURE 36

Linear Inequality in Two Variables

An inequality that can be written as

or

where A, B, and Care real numbers and Aand Bare not both 0, is a linear

inequality in two variables.

AxBy<C, AxBy◊C, AxBy>C, AxBy»C,

Consider the graph in FIGURE 36. The graph of the line divides the

points in the rectangular coordinate system into three sets:

1. Those points that lie on the line itself and satisfy

the equation like and

;

2. Those that lie in the half-plane above the line and

satisfy the inequality like and

;

3. Those that lie in the half-plane below the line and

satisfy the inequality like and

.

The graph of the line is called the boundary linefor the inequalities

and Graphs of linear inequalities in two variables are regions

in the real number plane that may or may not include boundary lines.

To graph a linear inequality in two variables, follow these steps.

x+y 75 x+y 6 5.

x+y= 5

1 - 3, - 124

xy< 53 1 0, 0 2

1 2, 4 24

xy> 53 1 5, 3 2

1 5, 0 24

xy 53 1 0, 5 2 , 1 2, 3 2 ,

x+y= 5

x

y

x + y = 5 (2, 4)

(0, 0)

(–3, –1)

(5, 3)

x + y > 5

x + y < 5

CAUTION When drawing the boundary line in Step 1, be careful to draw a solid

line if the inequality includes equality or a dashed line if equality is not

included. 16 , 72

1 ..., Ú 2

Graphing a Linear Inequality

Step 1 Draw the graph of the straight line that is the boundary.Make

the line solid if the inequality involves or. Make the line dashed

if the inequality involves or.

Step 2 Choose a test point.Choose any point not on the line, and substi-

tute the coordinates of that point in the inequality.

Step 3 Shade the appropriate region.Shade the region that includes the

test point if it satisfies the original inequality. Otherwise, shade the

region on the other side of the boundary line.

6 7

... Ú