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Vocabulary

Chapter 11 330

11-5 Solving Rational Equations

Review

1. What is on each side of an equal sign in a proportion? Explain.

_________________________________________________________________________

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2. Circle the property that you can use to solve any proportion.

Distributive Property Associative Property Cross Products Property

Vocabulary Builder

rational equation (noun) rash un ul ee kway zhun

Related Words: rational expression, rational function, extraneous solution

Definition: A rational equation is an equation that contains one or more rational expressions.

Examples: 34 1

1x2 5

12x and x 1 5

x 2 7 53

10

Nonexamples: 2x 1 y 5 5 and f (x) 5 3x 2 2 2x 1 1

Use Your Vocabulary

Write T for true or F for false.

3. The equation 3x 5

x4 is a rational equation.

4. The equation 17 2

19 5

1w is a rational equation.

5. The equation y 2 1

9 5 4 is a rational equation.

You can solve a rational equation by first multiplying each side of the equation by the least common denominator (LCD).

HSM12A1MC_1105_NL.indd 330 3/9/11 4:23:14 PM

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d.Problem 1

Problem 2

331 Lesson 11-5

Solving Equations With Rational Expressions

Got It? What is the solution of 13 1

3x 5

2x ? Check your solution.

6. The equation has been solved below. Write a justification for each step.

13 1

3x 5

2x

3x ? Q13 1

3xR 5 3x ? Q2

xR

3x ? Q13R 1 3x ? Q3

xR 5 3x ? Q2xR

x 1 9 5 6

x 5 23

7. Check to see whether 23 makes the equation true.

13 1

3 0 2

5

Solving by Factoring

Got It? What are the solutions of 5y 5

6y 2 2 6? Check your solutions.

8. Complete the steps to solve the equation.

?5y 5 ? Q 6

y 2 2 6R Multiply each side by the LCD, y2.

?5y 5 ? Q 6

y 2R 1 ? (26) Distributive Property

5 6 2 6y 2 Simplify.

6y 2 1 5y 2 6 5 0 Collect terms on one side.

(3y 2 ) (2y 1 ) 5 0 Factor the quadratic expression.

3y 2 5 0 or 2y 1 5 0 Zero-Product Property

y 5 or y 5 Solve for y.

9. Check your solutions in the original equation.

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Problem 4

Problem 3

Chapter 11 332

Solving a Work Problem

Got It? One hose can fill a pool in 12 h. Another hose can fill the same pool in 8 h. How long will it take for both hoses to fill the pool together?

10. Complete the model below to write an equation.

fraction of poolfirst hose can fill

in one hour

fraction of poolsecond hose can fill

in one hour

fraction of poolboth hoses can fill

in one hour

Let t = the time (in hours) it takes both hoses to fill the pool.Define

Write

Relate + =

= +1

12

1

HSM11_A1MC_1105_T91407

11. Solve the equation.

112 1

1 5

24t ? £1

11≥ 5 24t ? Q 1

t R Multiply each side by the LCD, 24t.

1 3t 5 Distributive Property

5 Simplify.

t 5 Solve for t.

12. It will take the two hoses about hours working together to fill the pool.

Solving a Rational Proportion

Got It? Find the solution(s) of 3b 1 2 5

5b 2 2. Check your solutions.

13. You can use the Cross Products Property.

3b 1 2 5

5b 2 2

? (b 2 2) 5 ? (b 1 2)

14. Now, solve the equation for b.

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Math Success

Now Iget it!

Need toreview

0 2 4 6 8 10

Lesson Check

Problem 5

1m

151 + =

2m

15=

m = 10

333 Lesson 11-5

Check off the vocabulary words that you understand.

rational equation extraneous solution Cross Products Property factor

Rate how well you can solve rational equations.

• Do you UNDERSTAND?

Error Analysis In the work shown at the right, what error did the student make in solving the rational equation?

18. Circle the first step for solving a rational equation with unlike denominators.

Find the LCD. Multiply the numerators. Factor.

19. What error did the student make?

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Checking to Find an Extraneous Solution

Got It? What is the solution of x 2 4x2 2 4

522

x 2 2? Check your solution.

15. Complete the steps for solving the equation or write a justification for a step.

(x 2 4)(x 2 2) 5 22(x2 2 4)

x2 2 1 8 5 22x2 1 Simplify each side of the equation.

3x2 2 5 0 Collect terms on one side.

3x(x 2 2) 5 0

3x 5 0 or 5 0 Zero-Product Property

x 5 0 or x 5 Solve for x.

16. Check your solutions in the original equation.

For x 5 0: 2

02 2 4

40

2

22

2 For x 5 :

2

22 2 4

40

2

22

2

2424 0 22 22 0 22

17. The solution is extraneous. The solution of the original equation is .

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