Example 1 Solve a Rational EquationExample 2 Elimination of a Possible SolutionExample 3 Work ProblemExample 4 Rate ProblemExample 5 Solve a Rational Inequality

Solve Check your solution.

The LCD for the three denominators is

Original equation

Multiply each side

by 24(3 – x).

1 1

11 1

6

Simplify.

Simplify.

Add.

Check Original equation

Simplify.

Simplify.

The solution is correct.

Answer: The solution is –45.

Answer:

Solve

Solve Check your solution.

The LCD is

Original equation

Multiply by the

LCD, (p2 – 1).

p – 1

1

1

1

DistributiveProperty

Simplify.

Simplify.

Add(2p2 – 2p + 1)to each side.

Factor.

orZero ProductProperty

Solve eachequation.

Divide eachside by 3.

Check Original equation

Simplify.

Simplify.

Since p = –1 results in a zero in the denominator, eliminate –1.

Answer: The solution is p = 2.

Simplify.

Original equation

Answer:

Solve

Mowing Lawns Tim and Ashley mow lawns together. Tim working alone could complete the job in 4.5 hours, and Ashley could complete it alone in 3.7 hours. How long does it take to complete the job when they work together?

In 1 hour, Tim could complete of the job.

In 1 hour, Ashley could complete of the job.

In t hours, Tim could complete or of the job.

In t hours, Ashley could complete or of the job.

Part completedby Tim plus

part completedby Ashley equals entire job.

1

Solve the equation.

Original equation

Multiply eachside by 16.65.

DistributiveProperty

Simplify.

Simplify.

Divide each side by 8.2.

Answer: It would take them about 2 hours working together.

Cleaning Libby and Nate clean together. Nate working alone could complete the job in 3 hours, and Libby could complete it alone in 5 hours. How long does it take to complete the job when they work together?

Answer: about 2 hours

Swimming Janine swims for 5 hours in a stream that has a current of 1 mile per hour. She leaves her dock and swims upstream for 2 miles and then back to her dock. What is her swimming speed in still water?

Words The formula that relates distance, time,

and rate is

Variables Let r be her speed in still water. Then her speed with the current is r + 1 and her speed against the current is r – 1.

Time going withthe current plus

time going againstthe current equals

totaltime.

5Equation

Solve the equation.

Originalequation

Multiply each

side by r2 – 1.

DistributiveProperty

r + 1 r – 1

1 1

Simplify.

Simplify.

Subtract 4r from each side.

Use the Quadratic Formula to solve for r.

Quadratic Formula

x = r, a = 5, b = –4, and c = –5

Simplify.

Simplify.

Use a calculator.

Answer: Since the speed must be positive, the answer is about 1.5 miles per hour.

Swimming Lynne swims for 1 hour in a stream that has a current of 2 miles per hour. She leaves her dock and swims upstream for 3 miles and then back to her dock. What is her swimming speed in still water?

Answer: about 6.6 mph

Solve

Step 1 Values that make the denominator equal to 0 are excluded from the denominator. For this inequality the excluded value is 0.

Step 2 Solve the related equation.

Related equation

Multiply each side by 9s.

Simplify.

Add.

Divide each side by 6.

Step 3 Draw vertical lines at the excluded value and at the solution to separate the number line into regions.

Now test a sample value in each region to determine if the values in the region satisfy the inequality.

Test

is a solution.

is not a solution.

Test

is a solution.

Test

Answer: The solution

Solve

Answer: