Chapter 7 Section 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Chapter Chapter 77Section Section 66

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley


Solving Equations with Rational Expressions

Distinguish between operations with rational expressions and equations with terms that are rational expressions.Solve equations with rational expressions.Solve a formula for a specified variable.

11

33

22

7.67.67.67.6


Objective 11

Distinguish between operations with rational expressions and equations with terms that are rational expressions.

Slide 7.6 - 3


Distinguish between operations with rational expressions and equations with terms that are rational.

Before solving equations with rational expressions, you must understand the difference between sums and differences of terms with rational coefficients, or rational expressions, and equations with terms that are rational expressions. Sums and differences are expressions to simplify. Equations are solved.

Slide 7.6 - 4

When adding or subtracting rational expressions, keep the LCD throughout the simplification.

When solving an equation, multiply each side by the LCD so the denominators are eliminated.


EXAMPLE 1

Solution:

Distinguishing between Expressions and Equations

Slide 7.6 - 5

Identify each of the following as an expression or an equation. Then simplify the expression or solve the equation.

5

2 3 6

x x

2 4

3 9

x x

equation expression

6 65

2 3 6

x x 3

3

3

2 4

9

x x

3 2 5x x 5x

2

9

x

6 4

9 9

x x

5


Objective 22

Solve equations with rational expressions.

Slide 7.6 - 6


Solve equations with rational expressions.

When an equation involves fractions, we use the multiplication property of equality to clear the fractions. Choose as multiplier the LCD of all denominators in the fractions of the equation.

Recall from Section 7.1 that the denominator of a rational expression cannot equal 0, since division by 0 is undefined. Therefore, when solving an equation with rational expressions that have variables in the denominator, the solution cannot be a number that makes the denominator equal 0.

Slide 7.6 - 7


Solve, and check the solution.

EXAMPLE 2

Solution:

Solving an Equation with Rational Expressions

Slide 7.6 - 8

2 3 6

5 3 5

m m

2 3 6

515 1

55

3

m m

6 9 5 18m m 9 189 9m

9m

Check:

1

2 3 6

5 35

91

55

9

2 3 6

5 3 5

m m

3 21 5 3 18

63 45 18 18 18

The use of the LCD here is different from its use in Section 7.5. Here, we use the multiplication property of equality to multiply each side of an equation by the LCD. Earlier, we used the fundamental property to multiply a fraction by another fraction that had the LCD as both its numerator and denominator.


While it is always a good idea to check solutions to guard against arithmetic and algebraic errors, it is essential to check proposed solutions when variables appear in denominators in the original equation.

Slide 7.6 - 9

Solve equations with rational expressions. (cont’d)

The steps used to solve an equation with rational expressions follow.

Step 1: Multiply each side of the equation by the LCD to clear the equation of fractions.

Step 2: Solve the resulting equation.

Step 3: Check each proposed solution by substituting it into the original equation. Reject any that cause a denominator to equal 0.


EXAMPLE 3

Solution:

2 21

1 1

x

x x


Slide 7.6 - 10

Solve, and check the proposed solution.

1

1 12 2

11

x

xx

xx

1 2 2x xx x 1 x

Since −1 causes the denominator to equal 0, the solution set is Ø.


EXAMPLE 4

Solution:2 2

2 3

2p p p p


Slide 7.6 - 11


2 66 6p 22 22 3 6pp pp

4p

2 3

22 1 2

11p p p

pp p p

p p p

The solution set is {4}.


EXAMPLE 5

Solution:2

8 3 3

4 1 2 1 2 1

r

r r r


Slide 7.6 - 12


2 1 2 1 2 18 3 3

2 1 2 1 2 1 2 12 1r r r

r

r r rr

r

8 6 3 6 3r r r 8 1212 12r rr r

4 0r 0r

Since 0 does not make any denominators equal 0, the solution set is {0}.


Solve, and check the proposed solution(s).

EXAMPLE 6

2

1 1 2

2 5 5 4x x


Slide 7.6 - 13

1 1 2

2 5 55 2 2 5

2 22 2x x x

x x xx

25 10 4 222x x 2 5 4 0x x

1 4 0x x

1 11 0x 4x 1x

Solution:

The solution set is {−4, −1}.

4 44 0x or


EXAMPLE 7

2

6 1 4

5 10 5 3 10a a a a


Slide 7.6 - 14


Solution:

6 1 4

5 2 55 2 5 5 2

2 55

a aa a

a aa a

6 30 5 10 20a a 40 40 04 20a

60a

The solution set is {60}.


Objective 33

Slide 7.6 - 15

Solve a formula for a specified variable.


EXAMPLE 8

for x

z yx y

Solving for a Specified Variable

Slide 7.6 - 16

Solve each formula for the specified variable.

Solution:

xz

xx

yx y y

z x y x

z z

xy

zx xx

xy x

z

2 1 1 for z

x y z

2 1 1

x yxy

zz xyz

2yz x

z z

z xy

2xy

y x xz

x

21 1xy

y xzxy xy

Remember that solving for a specified variable uses the same steps in the same sequence as solving an equation for the unknown.

1 2y x

z xy

2

xyz

y x

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Chapter 7 Section 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley