Intermediate Algebra (11th edition)

SECTION 9.7 Polynomial and Rational Inequalities 555

OBJECTIVE 2 Solve polynomial inequalities of degree 3 or greater.

Solving a Third-Degree Polynomial Inequality

Solve and graph the solution set of

This is a cubic(third-degree) inequality rather than a quadratic inequality, but it

can be solved using the preceding method by extending the zero-factor property to

more than two factors. (Step 1)

or or Zero-factor property

x= 1 or x= - 2 or x= 4 Solve each equation.

x- 1 = 0 x+ 2 = 0 x- 4 = 0

1 x- 121 x+ 221 x- 42 = 0

1 x- 121 x+ 221 x- 42 ...0.

EXAMPLE 4

Locate the numbers and 4 on a number line, as in FIGURE 21, to determine

the Intervals A, B, C, and D. (Step 2)

- 2, 1,

Set the factored

polynomial equalto 0.

Interval

A

Interval

B

Interval

C

Interval

D

T –2 F 1 T 4 F

FIGURE 21

–2 0 1 4

FIGURE 22

Interval Test Number Test of Inequality True or False?

A T

B0 F

C2 T

D5 28 ... 0 F

• 8 ... 0

8 ... 0

• 3 - 28 ... 0 We use a table to
  organize this
  information. ( Verify it.)

Substitute a test number from each interval in the originalinequality to deter-

mine which intervals satisfy the inequality. (Step 3)

The numbers in Intervals A and C are in the solution set, which is written in interval

notation as and graphed in FIGURE 22. The three endpoints are

included since the inequality symbol, , includes equality. (Step 4)...

1 - q, - 24 ́ 3 1, 4 4 ,

NOW TRY

OBJECTIVE 3 Solve rational inequalities. Inequalities that involve rational

expressions, called rational inequalities,are solved similarly using the following steps.

Solving a Rational Inequality

Step 1 Write the inequality so that 0 is on one sideand there is a single

fraction on the other side.

Step 2 Determine the numbers that make the numerator or denomina-

tor equal to 0.

Step 3 Divide a number line into intervals.Use the numbers from Step 2.

Step 4 Find the intervals that satisfy the inequality.Test a number from

each interval by substituting it into the originalinequality.

Step 5 Consider the endpoints separately.Exclude any values that make

the denominator 0.

NOW TRY
EXERCISE 4
Solve and graph the solution
set.

1 x+ 421 x- 3212 x+ 12 ... 0

–4 –^1203

NOW TRY ANSWER

4. 1 - q, - 44 ́C-^12 , 3D