Beginning Algebra, 11th Edition

7.6 Solving Equations with Rational

Expressions

Solving Equations with Rational Expressions

Step 1 Multiply each side of the equation by the LCD
to clear the equation of fractions. Be sure to
distribute to everyterm on bothsides.

Step 2 Solve the resulting equation.

Step 3 Check each proposed solution.

Solve.

Factor.

The LCD is Note that 3 and cannot be solutions,

as they cause a denominator to equal 0.

Multiply by the LCD.

Distributive property

Distributive property

Standard form

Factor.

or Zero-factor property

Reject or Solve for x.

Since 3 causes denominators to equal 0, the only solution is

Thus, 5 - 106 is the solution set.

x= 3 x=- 10

x- 3 = 0 x+ 10 = 0

1 x- 321 x+ 102 = 0

x^2 + 7 x- 30 = 0

x^2 + 3 x+ 4 x- 12 = 18

x 1 x+ 32 + 41 x- 32 = 18

= 1 x- 321 x+ 32

18

1 x- 321 x+ 32

1 x- 321 x+ 32 a

x

x- 3

+

4

x+ 3

b

1 x- 321 x+ 32. - 3

x

x- 3

+

4

x+ 3

=

18

1 x- 321 x+ 32

x

x- 3

+

4

x+ 3

=

18

x^2 - 9

CONCEPTS EXAMPLES

7.7 Applications of Rational Expressions

Solving Problems about Distance, Rate, and Time
Use the formulas relating d, r, and t.

Solving Problems about Work

Step 1 Read the problem carefully.

Step 2 Assign a variable. State what the variable
represents. Put the information from the
problem into a table. If a job is done in tunits
of time, the rate is

Step 3 Write an equation. The sum of the fractional
parts should equal 1 (whole job).

Step 4 Solve the equation.

Steps 5 State the answer and check the solution.
and 6

1

t^.

drt, r

d

t

, t

d

r

It takes the regular mail carrier 6 hr to cover her route. A substitute

takes 8 hr to cover the same route. How long would it take them to

cover the route together?

Let x=the number of hours required to cover the route together.

The LCD is 24.

Distributive property

Combine like terms.

Divide by 7.

It would take them or hr, to cover the route together.

The solution checks because^16 A^247 B+^18 A^247 B =1.

(^247) hr, (^3 37)
x=
24
7
7 x= 24
4 x+ 3 x= 24
24 a
1
6
x+
1
8
xb= 24112
1
6
x+
1
8
x= 1
(continued)
486 CHAPTER 7 Rational Expressions and Applications
Part of the
Rate Time Job Done
Regular x
Substitute^18 x^18 x
1
6 x
1
6
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