Intermediate Algebra (11th edition)

Graphing a Linear Inequality

Graph

First graph the boundary line, shown in FIGURE 39. The points of the boundary

line do not belong to the inequality (because the inequality symbol is

not ). For this reason, the line is dashed. Now solve the inequality for y.

Subtract x.

Multiply by. Change

Because of the is greater thansymbol that occurs when

the inequality is solved for y,shade abovethe line.

CHECK Choose a test point not on the line, say,.

Let and.

✓ True

This result agrees with the decision to shade above the

line. The solution set, graphed in FIGURE 39, includes

only those points in the shaded half-plane (not those

on the line).

06 4

0 - 31026 x= 0 y= 0

?

4

x- 3 y 6 4

1 0, 0 2

y 7 -^136 to 7.

1

3

x-

4

3

- 3 y6-x+ 4

x- 3 y 6 4

...

x- 3 y 6 4 6 ,

x- 3 y 6 4.

EXAMPLE 2

SECTION 3.4 Linear Inequalities in Two Variables 177

In practice, the graphs in FIGURES 40(a) AND (b)are graphed on the same axes.

CHECK Using FIGURE 40(c), choose a test point from each of the four regions

formed by the intersection of the boundary lines. Verify that only ordered pairs in the

heavily shaded region satisfy bothinequalities. NOW TRY

NOW TRY
EXERCISE 2
Graph. 3 x-y 66

x

y

(0, 0)

0

2

4

x – 3y < 4

x – 3 y = 4

FIGURE 39

OBJECTIVE 2 Graph the intersection of two linear inequalities. A pair of

inequalities joined with the word andis interpreted as the intersection of the solution

sets of the inequalities. The graph of the intersection of two or more inequalities is

the region of the plane where all points satisfy all of the inequalities at the same time.

Graphing the Intersection of Two Inequalities

Graph and

To begin, we graph each of the two inequalities and sepa-

rately, as shown in FIGURES 40(a) AND (b). Then we use heavy shading to identify the

intersection of the graphs, as shown in FIGURE 40(c).

2 x+ 4 yÚ 5 xÚ 1

2 x+ 4 yÚ 5 xÚ1.

EXAMPLE 3

x

y

0 5

2

5

4

2 x + 4y ≥ 5

x

y

–2 0

2

–2 x ≥ 1

x

y

0

2 x + 4y ≥ 5

and x ≥ 1

(a) (b) (c)
FIGURE 40
NOW TRY ANSWERS

2. 
   
   
   
   3. 
      
   
   
   
   

–6

y

3 x – y < 6

0

x

2

NOW TRY

NOW TRY
EXERCISE 3
Graph and .x+y 63 y... 2

x

y

0 3

3

2

x + y < 3

and y Ä 2