Intermediate Algebra (11th edition)

SECTION 9.7 Polynomial and Rational Inequalities 557

Solving a Rational Inequality

Solve and graph the solution set of

Write the inequality so that 0 is on one side. (Step 1)

Subtract 2.

Use as the common denominator.

Write as a single fraction.

Combine like terms in the numerator.

- x- 6

x+ 2

... 0

x- 2 - 2 x- 4

x+ 2

... 0

x+ 2

x- 2

x+ 2

-

21 x+ 22

x+ 2

... 0

x- 2

x+ 2

- 2 ... 0

x- 2

x+ 2

... 2.

EXAMPLE 6

Be careful

with signs.

NOW TRY
EXERCISE 6
Solve and graph the solution
set.

x- 3

x+ 3

... 2

–9 –3 0

–6 –2 0

FIGURE 25

The number makes the numerator 0, and makes the denominator 0. (Step 2)

These two numbers determine three intervals. (Step 3) Test one number from each in-

terval (Step 4) to see that the solution set is

The number satisfies the original inequality, but does not since it makes the

denominator 0. (Step 5) FIGURE 25shows a graph of the solution set.

- 6 - 2

1 - q, - 64 ́ 1 - 2, q 2.

- 6 - 2

NOW TRY

4.Concept Check The solution set of the inequality is the interval

. Without actually performing any work, give the solution set of the inequality
x^2 +x- 12 Ú 0.

1 - 4, 3 2

x^2 +x- 1260

NOW TRY ANSWER

6. 1 - q, - 94 ́ 1 - 3, q 2

x

y

0

13

3

f(x) = x^2 – 4x + 3

x

y

0

–4

–8

f(x) = 3x^2 + 10x – 8

2

3 x

y

0

–2 5

10

f(x) = –x^2 + 3x + 10

Complete solution available
on the Video Resources on DVD

In Exercises 1– 3, the graph of a quadratic function ƒ is given. Use the graph to find the solu-

tion set of each equation or inequality. See Example 1.

1. (a)
   (b)
   (c)x^2 - 4 x+ 360

x^2 - 4 x+ 370

x^2 - 4 x+ 3 = 0 2. (a)

(b)

(c) 3 x^2 + 10 x- 860

3 x^2 + 10 x- 8 Ú 0

3 x^2 + 10 x- 8 = 0 3. (a)

(b)

(c) -x^2 + 3 x+ 10 ... 0

• x^2 + 3 x+ 10 Ú 0


• x^2 + 3 x+ 10 = 0

9.7 EXERCISES