Intermediate Algebra (11th edition)

Solving a Rational Inequality

Solve and graph the solution set of

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

Subtract 1.

Use as the common denominator.

Write the left side as a single fraction.

Combine like terms in the numerator.

- x+ 2

x- 3

70

- 1 - x+ 3

x- 3

70

x- 3

- 1

x- 3

-

x- 3

x- 3

70

- 1

x- 3

- 170

- 1

x- 3

7 1.

EXAMPLE 5

556 CHAPTER 9 Quadratic Equations, Inequalities, and Functions

CAUTION When solving a rational inequality, any number that makes the

denominator 0 must be excluded from the solution set.

Be careful

with signs.

F 23 T F

Interval

C

Interval

B

Interval

A

FIGURE 23

031452

FIGURE 24

Interval Test Number Test of Inequality True or False?

A0 F

B 2.5 T

C4 - 171 F

271

1

371

The sign of will change from positive to negative or negative to positive

only at those numbers that make the numerator or denominator 0. The number 2

makes the numerator 0, and 3 makes the denominator 0. (Step 2) These two numbers,

2 and 3, divide a number line into three intervals. See FIGURE 23. (Step 3)

• x+ 2
  x- 3

Testing a number from each interval in the originalinequality, gives

the results shown in the table. (Step 4)

• 1

x- 3 7 1,

The solution set is the interval This interval does not include 3 since it would

make the denominator of the original equality 0. The number 2 is not included either

since the inequality symbol, , does not include equality. (Step 5) See 7 FIGURE 24.

1 2, 3 2.

NOW TRY

NOW TRY
EXERCISE 5
Solve and graph the solution
set.

3

x+ 1

74

NOW TRY ANSWER

5. A-1, -^14 B

–1 –^140