Intermediate Algebra (11th edition)

7.4 Equations with Rational Expressions

and Graphs

Solving an Equation with Rational Expressions

Step 1 Determine the domain of the variable.

Step 2 Multiply each side of the equation by the least
common denominator.

Step 3 Solve the resulting equation.

Step 4 Check that each proposed solution is in the
domain, and discard any values that are not.
Check the remaining proposed solutions in the
original equation.

Graphing a Rational Function
The graph of a rational function (written in lowest terms)
may have one or more breaks. At such points, the graph
will approach an asymptote.

Solve.

Note that 0 and 2 are

excluded from the domain.

3 x+ 2

x- 2

+

2

x 1 x- 22

=

- 1

x

CONCEPTS EXAMPLES

Multiply by the LCD,

Distributive property

Add x. Subtract 2.

Factor.

or Zero-factor property

or Solve each equation.

Of the two proposed solutions, 0 must be discarded because it is not

in the domain. The solution set is.

x

y

Vertical

asymptote

x = –2

f(x) =x + 2^1

0

–4

4

–1 1

–5–4–3

Horizontal

asymptote

y = 0

5 - 16

x= 0 x=- 1

3 x= 0 x+ 1 = 0

3 x 1 x+ 12 = 0

3 x^2 + 3 x= 0

3 x^2 + 2 x+ 2 =-x+ 2

x 13 x+ 22 + 2 =- 1 x- 22 x 1 x- 22.

7.5 Applications of Rational Expressions

To solve a motion problem, use the formula

or one of its equivalents,

or

To solve a work problem, use the fact that if a complete
job is done in tunits of time, the rate of work is job per
unit of time.

1

t

r

d

t

t.

d

r

drt

Solve.

A canal has a current of 2 mph. Find the rate of Amy’s boat in still

water if it goes 11 mi downstream in the same time that it goes 8 mi

upstream.

Let xrepresent the rate of the boat in still water.

Distance Rate Time

Downstream 11

Upstream 8 8

x- 2

x- 2

11

x+ (^2) x+ 2 Times
are equal.
Write an equation.
The LCD is
Multiply by the LCD.
Distributive property
Subtract 8x. Add 22.
Divide by 3.
The rate in still water is 12 23 mph.
x= 12

2

3

3 x= 38

11 x- 22 = 8 x+ 16

111 x- 22 = 81 x+ 22

1 x+ 221 x- 22.

11

x+ 2

=

8

x- 2

(continued)

CHAPTER 7 Summary 419