• rational equation

• weighted average

• rational inequality

Solve a Rational Equation

The LCD for the terms is 24(3 – x).

Original equation

Solve . Check your solution.

Multiply each side by 24(3 – x).

Solve a Rational Equation

Distributive Property

Simplify.

Simplify.

Add 6x and –63 to each side.

Solve a Rational Equation

Original equationCheck

x = –45

Simplify.

Simplify.

The solution is correct.

Solve a Rational Equation

Answer: The solution is –45.

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

Solve .

A. –2

B.

C.

D. 2

Solve a Rational Equation

The LCD is (p + 1)(p – 1).

Original equation

Solve Check your solution.

Multiply by the LCD.

Solve a Rational Equation

(p – 1)(p2 – p – 5) = (p2 – 7)(p + 1) + p(p + 1)(p – 1)

p3 – p2 – 5p – p2 + p + 5 = p3 + p2 – 7p – 7 + p3 – p

p3 – 2p2 – 4p + 5 = 2p3 + p2 – 8p – 7

0 = p3 + 3p2 – 4p – 12

Divide commonfactors.

DistributiveProperty

Simplify.

Subtract p3 – 2p2 – 4p + 5from each side.

Solve a Rational Equation

Zero Product

Property

0 = (p + 3)(p + 2)(p – 2)

Factor.

0 = p + 3 or 0 = p + 2 or 0 = p – 2

Original equation

Check Try p = –3.

p = –3

Solve a Rational Equation

Simplify.

Simplify.

Original equation

Try p = –2.

Solve a Rational Equation

Simplify.

p = –2

Simplify.

Solve a Rational Equation

Answer: The solutions are –3, –2 and 2.

Try p = 2.

Original equation

Simplify.

p = 2

Simplify.

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. 4

B. –2

C. 2

D. –4

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 asolution needs to be added to anoriginal solution to create a newsolution.

Mixture Problem

Plan Each solution has a certainpercentage that is salt. Thepercentage of brine in the finalsolution must equal the amount ofbrine divided by the total solution.

Percentage of brine in solution

Mixture Problem

Substitute.

Simplify numerator.

LCD is 100(10 + x).

Solve Write a proportion.

Mixture Problem

Distribute.

Subtract 50x and160.

Divide each side by 30.

Answer: Aaron needs to add ounces of 80% brine solution.

Simplify.

Divide common factors.

Mixture Problem

Simplify.

Check Originalequation

?

?

0.5 = 0.5 Simplify.

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

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 20% base. How much of the solution should be added to create a solution that is 40% base?

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, andthe 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

Solve

Original equation

Time going withthe current plus

time going againstthe current equals

totaltime.

5

Let r equal her speed in still water. Thenher speed with the current is r + 1, andher speed against the current is r – 1.

Distance Problem

Divide Common Factors

Distribute.

Simplify.

Subtract 4rfrom each side.

(r + 1)2 + (r – 1)2 = 5(r2 – 1) Simplify.

Multiply each side by r2 – 1.

Distance Problem

Use the Quadratic Formula to solve for r.

Quadratic Formula

x = r, a = 5, b = – 4, and c = –5

Simplify.

Simplify.

Distance Problem

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

Check Original equation

r ≈ 1.5 or –0.7 Use a calculator.

r = 1.5?

Simplify.

Simplify.?

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

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 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?

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 We are given how long it takes Wuyiand Uima working alone to mow aparticular lawn. We need to determinehow long it would take them together.

Plan Wuyi can mow the lawn in 4.5 hours,

so the rate of mowing is of a

lawn per hour.

Work Problems

Uima can mow the lawn in 3.7 hours,

so the rate of mowing is of a lawn

per hour.

The combined rate is

Work Problems

Multiply both sidesby x.

Solve

Write the equation.

Add

Answer: It would take Wuyi and Uima about 2 hours to mow the lawn together.

x ≈ 2.0304 Multiply 1 by

Work Problems

x ≈ 2

Simplify.

Check

Originalequation

?

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

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 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?

Solve a Rational Inequality

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

Related equation

Step 2 Solve the related equation.

Solve

Solve a Rational Inequality

Multiply each side by 9k.

Simplify.

Add.

Divide each side by 6.

Solve a Rational Inequality

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.

Solve a Rational Inequality

Test k = –1.

k < 0 is a solution.

Solve a Rational Inequality

Test k = .

0 < k < is not a solution.

Solve a Rational Inequality

Test k = 1.

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. x < 0

B. x > 0

C. x < 0 or x > 4

D. 0 < x < 4

Solve .