You simplified rational expressions Solve rational equations rational

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You simplified rational expressions. • Solve rational equations.

You simplified rational expressions. • Solve rational equations.

 • rational equation • weighted average

• rational equation • weighted average

Solve a Rational Equation Solve . Check your solution. The LCD for the terms

Solve a Rational Equation Solve . Check your solution. The LCD for the terms is 24(3 – x). Original equation Multiply each side by 24(3 – x).

Solve a Rational Equation Distributive Property Simplify. Add 6 x and – 63 to

Solve a Rational Equation Distributive Property Simplify. Add 6 x and – 63 to each side.

Solve a Rational Equation Original equation Check x = – 45 Simplify. The solution

Solve a Rational Equation Original equation Check x = – 45 Simplify. The solution is correct.

Solve a Rational Equation Answer:

Solve a Rational Equation Answer:

Solve a Rational Equation Answer: The solution is – 45.

Solve a Rational Equation Answer: The solution is – 45.

Solve A. – 2 B. C. D. 2 .

Solve A. – 2 B. C. D. 2 .

Solve A. – 2 B. C. D. 2 .

Solve A. – 2 B. C. D. 2 .

Solve a Rational Equation Solve Check your solution. The LCD is (p + 1)(p

Solve a Rational Equation Solve Check your solution. The LCD is (p + 1)(p – 1). Original equation Multiply by the LCD.

Solve a Rational Equation (p – 1)(p 2 – p – 5) = (p

Solve a Rational Equation (p – 1)(p 2 – p – 5) = (p 2 – 7)(p + 1) + p(p + 1)(p – 1) Divide common factors. p 3 – p 2 – 5 p – p 2 + p + 5 = p 3 + p 2 – 7 p – 7 + p 3 – p Distributive Property p 3 – 2 p 2 – 4 p + 5 = 2 p 3 + p 2 – 8 p – 7 Simplify. 0 = p 3 + 3 p 2 – 4 p – 12 Subtract p 3 – 2 p 2 – 4 p + 5 from each side.

Solve a Rational Equation 0 = (p + 3)(p + 2)(p – 2) Factor.

Solve a Rational Equation 0 = (p + 3)(p + 2)(p – 2) Factor. 0 = p + 3 or 0 = p + 2 or 0 = p – 2 Zero Product Property Check Try p = – 3. Original equation ? p = – 3

Solve a Rational Equation ? Simplify. or Try p = – 2. Original equation

Solve a Rational Equation ? Simplify. or Try p = – 2. Original equation

Solve a Rational Equation ? p = – 2 ? Simplify.

Solve a Rational Equation ? p = – 2 ? Simplify.

Solve a Rational Equation Try p = 2. Original equation ? p=2 ? Simplify.

Solve a Rational Equation Try p = 2. Original equation ? p=2 ? Simplify. Answer:

Solve a Rational Equation Try p = 2. Original equation ? p=2 ? Simplify.

Solve a Rational Equation Try p = 2. Original equation ? p=2 ? Simplify. Answer: The solutions are – 3, – 2 and 2.

Mixture Problem BRINE Aaron adds an 80% brine (salt and water) solution to 16

Mixture Problem BRINE Aaron adds an 80% brine (salt and water) solution to 16 ounces of solution that is 10% brine. How much of the solution should be added to create a solution that is 50% brine? Understand Aaron needs to know how much of a solution needs to be added to an original solution to create a new solution.

Mixture Problem Plan Each solution has a certain percentage that is salt. The percentage

Mixture Problem Plan Each solution has a certain percentage that is salt. The percentage of brine in the final solution must equal the amount of brine divided by the total solution. Percentage of brine in solution

Mixture Problem Solve Write a proportion. Substitute. Simplify numerator. LCD is 100(16 + x).

Mixture Problem Solve Write a proportion. Substitute. Simplify numerator. LCD is 100(16 + x).

Mixture Problem Divide common factors. Simplify. Distribute. Subtract 50 x and 160. Divide each

Mixture Problem Divide common factors. Simplify. Distribute. Subtract 50 x and 160. Divide each side by 30. Answer:

Mixture Problem Divide common factors. Simplify. Distribute. Subtract 50 x and 160. Divide each

Mixture Problem Divide common factors. Simplify. Distribute. Subtract 50 x and 160. Divide each side by 30. Answer: Aaron needs to add brine solution. ounces of 80%

Mixture Problem Check Original equation ? ? 0. 5 = 0. 5 Simplify.

Mixture Problem Check Original equation ? ? 0. 5 = 0. 5 Simplify.

BRINE Janna adds a 65% base solution to 13 ounces of solution that is

BRINE Janna adds a 65% base solution to 13 ounces of solution that is 20% base. How much of the solution should be added to create a solution that is 40% base? A. 9. 6 ounces B. 10. 4 ounces C. 11. 8 ounces D. 12. 3 ounces

BRINE Janna adds a 65% base solution to 13 ounces of solution that is

BRINE Janna adds a 65% base solution to 13 ounces of solution that is 20% base. How much of the solution should be added to create a solution that is 40% base? A. 9. 6 ounces B. 10. 4 ounces C. 11. 8 ounces D. 12. 3 ounces

Distance Problem SWIMMING Lilia swims for 5 hours in a stream that has a

Distance Problem SWIMMING Lilia 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? Understand We are given the speed of the current, the distance she swims upstream, and the total time. Plan She swam 2 miles upstream against the current and 2 miles back to the dock with the current. The formula that relates distance, time, and rate is d = rt or

Distance Problem Let r equal her speed in still water. Then her speed with

Distance Problem Let r equal 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 total equals time. 5 Solve Original equation

Distance Problem Multiply each side by r 2 – 1. Divide Common Factors (r

Distance Problem Multiply each side by r 2 – 1. Divide Common Factors (r + 1)2 + (r – 1)2 = 5(r 2 – 1) Simplify. Distribute. Simplify. Subtract 4 r from each side.

Distance Problem Use the Quadratic Formula to solve for r. Quadratic Formula x =

Distance Problem Use the Quadratic Formula to solve for r. Quadratic Formula x = r, a = 5, b = – 4, and c = – 5 Simplify.

Distance Problem r ≈ 1. 5 or – 0. 7 Answer: Use a calculator.

Distance Problem r ≈ 1. 5 or – 0. 7 Answer: Use a calculator.

Distance Problem r ≈ 1. 5 or – 0. 7 Use a calculator. Answer:

Distance Problem r ≈ 1. 5 or – 0. 7 Use a calculator. Answer: Since speed must be positive, the answer is about 1. 5 miles per hour. Original equation Check ? r = 1. 5 ? Simplify.

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? A. about 0. 6 mph B. about 2. 0 mph C. about 4. 6 mph D. about 6. 6 mph

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? A. about 0. 6 mph B. about 2. 0 mph C. about 4. 6 mph D. about 6. 6 mph

Work Problems MOWING LAWNS Wuyi and Uima mow lawns together. Wuyi working alone could

Work Problems MOWING LAWNS Wuyi and Uima mow lawns together. Wuyi working alone could complete a particular job in 4. 5 hours, and Uima could complete it alone in 3. 7 hours. How long does it take to complete the job when they work together? Understand Plan We are given how long it takes Wuyi and Uima working alone to mow a particular lawn. We need to determine how long it would take them together. Wuyi can mow the lawn in 4. 5 hours, so the rate of mowing is lawn per hour. of a

Work Problems Uima can mow the lawn in 3. 7 hours, so the rate

Work Problems Uima can mow the lawn in 3. 7 hours, so the rate of mowing is per hour. The combined rate is of a lawn

Work Problems Solve Write the equation. Add Multiply both sides by x. x ≈

Work Problems Solve Write the equation. Add Multiply both sides by x. x ≈ 2. 0304 Answer: Multiply 1 by

Work Problems Solve Write the equation. Add Multiply both sides by x. x ≈

Work Problems Solve Write the equation. Add Multiply both sides by x. x ≈ 2. 0304 Multiply 1 by Answer: It would take Wuyi and Uima about 2 hours to mow the lawn together.

Work Problems Check Original equation ? x≈2 Simplify.

Work Problems Check Original equation ? x≈2 Simplify.

PAINTING Adriana and Monique paint rooms together. Adriana working alone could complete a particular

PAINTING Adriana and Monique paint rooms together. Adriana working alone could complete a particular job in 6. 4 hours, and Monique could complete it alone in 4. 8 hours. How long does it take to complete the job when they work together? A. about 2 hours and 28 minutes B. about 2 hours and 36 minutes C. about 2 hours and 45 minutes D. about 2 hours and 56 minutes

PAINTING Adriana and Monique paint rooms together. Adriana working alone could complete a particular

PAINTING Adriana and Monique paint rooms together. Adriana working alone could complete a particular job in 6. 4 hours, and Monique could complete it alone in 4. 8 hours. How long does it take to complete the job when they work together? A. about 2 hours and 28 minutes B. about 2 hours and 36 minutes C. about 2 hours and 45 minutes D. about 2 hours and 56 minutes