example 1 solve a rational equation the lcd for the terms is 24(3 – x). original equation solve....

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.

Answer: The solution is –45.

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 – pp3 – 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 ProductProperty

0 = (p + 3)(p + 2)(p – 2)Factor.

0 = p + 3 or 0 = p + 2 or 0 = p – 2

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

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(16 + 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.

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 thecurrent and 2 miles back to the dock withthe 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 4r from 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.