Example 1 Example 2 Example 3 Example 4

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Example 1 Example 2 Example 3 Example 4 Example 5 Solve a Rational Equation

Example 1 Example 2 Example 3 Example 4 Example 5 Solve a Rational Equation Elimination of a Possible Solution Work Problem Rate Problem Solve a Rational Inequality

Solve Check your solution. The LCD for the three denominators is Original equation Multiply

Solve Check your solution. The LCD for the three denominators is Original equation Multiply each side by 24(3 – x).

6 1 1 1 Simplify. Add.

6 1 1 1 Simplify. Add.

Check Original equation Simplify. The solution is correct.

Check Original equation Simplify. The solution is correct.

Answer: The solution is – 45.

Answer: The solution is – 45.

Solve Answer:

Solve Answer:

Solve Check your solution. The LCD is Original equation p– 1 1 Multiply by

Solve Check your solution. The LCD is Original equation p– 1 1 Multiply by the LCD, (p 2 – 1).

Distributive Property Simplify. Add (2 p 2 – 2 p + 1) to each

Distributive Property Simplify. Add (2 p 2 – 2 p + 1) to each side.

Divide each side by 3. Factor. or Zero Product Property Solve each equation.

Divide each side by 3. Factor. or Zero Product Property Solve each equation.

Check Original equation Simplify.

Check Original equation Simplify.

Original equation Simplify. Since p = – 1 results in a zero in the

Original equation Simplify. Since p = – 1 results in a zero in the denominator, eliminate – 1. Answer: The solution is p = 2.

Solve Answer:

Solve Answer:

Mowing Lawns Tim and Ashley mow lawns together. Tim working alone could complete the

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 In 1 hour, Ashley could complete of the job.

In t hours, Tim could complete or In t hours, Ashley could complete Part

In t hours, Tim could complete or In t hours, Ashley could complete Part completed by Tim plus of the job. or part completed by Ashley equals of the job. entire job. 1

Solve the equation. Original equation Multiply each side by 16. 65. Distributive Property Simplify.

Solve the equation. Original equation Multiply each side by 16. 65. Distributive Property Simplify.

Simplify. Divide each side by 8. 2. Answer: It would take them about 2

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

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

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 with the current plus time going against the current equals total time.

Time going with the current plus time going against the current equals total time. 5 Equation Solve the equation. Original equation

Multiply each side by r 2 – 1. r+1 r– 1 1 Distributive Property

Multiply each side by r 2 – 1. r+1 r– 1 1 Distributive Property 1 Simplify. Subtract 4 r from each side.

Use the Quadratic Formula to solve for r. Quadratic Formula x = r, a

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

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

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

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 9 s. Simplify. Add. Divide each side by 6.

Multiply each side by 9 s. Simplify. Add. Divide each side by 6.

Step 3 Draw vertical lines at the excluded value and at the solution to

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.

Test is a solution.

Test is not a solution.

Test is not a solution.

Test is a solution.

Test is a solution.

Answer: The solution

Answer: The solution

Solve Answer:

Solve Answer:

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