Beginning Algebra, 11th Edition

OBJECTIVE 1 Graph linear inequalities in two variables.Consider the

graph in FIGURE 35. The graph of the line divides the points in the rectan-

gular 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 region above the line and satisfy the inequality

like and

3.Those that lie in the region 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.

Graphing a Linear Inequality

Graph

The inequality means that

We begin by graphing the equation a line with intercepts and

as shown in FIGURE 36. This boundary line divides the plane into two regions,

one of which satisfies the inequality. A test pointgives a quick way to find the cor-

rect region. We choose any point noton the boundary line and substitute it into the

given inequality to see whether the resulting statement is true or false. The point

is a convenient choice.

Original inequality

Let and

True

Since the last statement is true, we shade the region that includes the test point

See FIGURE 36. The shaded region, along with the boundary line, is the

desired graph.

1 0, 0 2.

0 ... 6

0 + 0 ...

?

6

2102 + 3102 ... x= 0 y=0.

?

6

2 x+ 3 y... 6

1 0, 0 2

1 3, 0 2 ,

2 x+ 3 y= 6, 1 0, 2 2

2 x+ 3 y 66 or 2 x+ 3 y= 6.

2 x+ 3 y... 6

2 x+ 3 y...6.

EXAMPLE 1

x+y 75 x+ y 6 5.

x+ y= 5

3 1 0, 0 2 1 - 3 - 124

x+y 65

3 1 5, 3 2 1 2, 4 24

x+y 75

1 0, 5 2 , 1 2, 3 2 , 1 5, 0 24

xy 53

x+ y= 5

224 CHAPTER 3 Linear Equations and Inequalities in Two Variables; Functions

NOW TRY
EXERCISE 1
Graph x+ 3 y...6.

x

y

x + y = 5 (2, 4)

(0, 0)

(–3, –1)

(5, 3)

x + y > 5

x + y < 5

FIGURE 35

Use as

a test point.

1 0, 0 2

(0, 2)

(3, 0)

yy

x

2 x 3 y 6

Boundary

line

Test

point

2 x 3 y 6

(0, 0)

FIGURE 36 NOW TRY

NOW TRY ANSWER

1. 
   

x

y

0

x + 3y Ä 6

2

6

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