Intermediate Algebra (11th edition)

9.7 Polynomial and Rational Inequalities

Solving a Quadratic (or Higher-Degree Polynomial)
Inequality

Step 1 Write the inequality as an equation and solve.

Step 2 Use the numbers found in Step 1 to divide a
number line into intervals.

Step 3 Substitute a test number from each interval into
the original inequality to determine the
intervals that belong to the solution set.

Step 4 Consider the endpoints separately.

Solving a Rational Inequality

Step 1 Write the inequality so that 0 is on one side and
there is a single fraction on the other side.

Step 2 Determine the numbers that make the numerator
or denominator 0.

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

Step 4 Substitute a test number from each interval into
the original inequality to determine the intervals
that belong to the solution set.

Step 5 Consider the endpoints separately.

CONCEPTS EXAMPLES

Solve

or

Intervals:

Test values:

makes the original inequality false, makes it true,

and makes it false. Choose the interval(s) which yield(s) a true

statement. The solution set is the interval

Solve

Subtract 4.

Write with a

common denominator.

Subtract fractions.

makes the numerator 0, and makes the denominator 0.

from A makes the original inequality false, from B makes it

true, and 0 from C makes it false.

The solution set is the interval The endpoint is not

included since it makes the denominator 0.

C-^83 , - 2 B. - 2

- 4 -^73

F T F

AB C

(^8) –2

• 3

-^83 - 2

• 3 x- 8
  x+ 2

Ú 0

x

x+ 2

-

41 x+ 22

x+ 2

Ú 0

x

x+ 2

- 4 Ú 0

x

x+ 2

Ú4.

A-2, -^12 B.

x= 0

x=- 3 x=- 1

- 3, -1, 0

A-2, -

1

2 B, A-

1

2 , qB

1 - q, - 22 ,

–3 –2 0

F

A

F

C

T

B

–1^1

• 2

x=-^12 x=- 2

12 x+ 121 x+ 22 = 0

2 x^2 + 5 x+ 2 = 0

2 x^2 + 5 x+ 26 0.

CHAPTER 9

9.1 Solve each equation by using the square root property or completing the square.

1. 2. 3.

*4. 13 x- 222 =- 25 5.x^2 + 4 x= 15 6. 2 x^2 - 3 x=- 1

t^2 = 121 p^2 = 3 12 x+ 522 = 100

*This exercise requires knowledge of complex numbers.

562 CHAPTER 9 Quadratic Equations, Inequalities, and Functions

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