Beginning Algebra, 11th Edition

The ideas of Example 1can be summarized as follows.

SECTION 7.6 Solving Equations with Rational Expressions 457

Uses of the LCD

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

simplification. (See Example 1(a).)

When solving an equation, multiply each side by the LCD so that denominators

are eliminated. (See Example 1(b).)

Solving an Equation with Rational Expressions

Solve, and check the solution.

Multiply each side by the LCD, 6.

Distributive property

Distributive property

Multiply.

Combine like terms.

p= 4 Subtract 2.

p+ 2 = 6

3 p- 2 p+ 2 = 6

3 p- 21 p 2 - 21 - 12 = 6

3 p- 21 p- 12 = 6

6 a

p

2

b - 6 a

p- 1

3

b = 6112

6 a

p

2

-

p- 1

3

b = 6112

p

2

-

p- 1

3

= 1

NOW TRY EXAMPLE 2
EXERCISE 2
Solve, and check the solution.

x+ 5

5

• x
  7

=

3

7

NOW TRY ANSWER

2. 5 - 106

Use parentheses around

p- 1 to avoid errors.

Be careful

with signs.

Check to see that 546 is the solution set by replacing pwith 4 in the original equation.

CAUTION In Example 2,we used the multiplication property of equality to

multiply each side of an equationby the LCD. In Section 7.5,we used the funda-

mental property to multiply a fractionby another fraction that had the LCD as

both its numerator and denominator. Be careful not to confuse these procedures.

Recall from Section 7.1that 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.

A value of the variable that appears to be a solution after both sides of a rational

equation are multiplied by a variable expression is called a proposed solution.All

proposed solutions must be checked in the original equation.

OBJECTIVE 2 Solve equations with rational expressions.When an equa-

tion involves fractions, as in Example 1(b),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.

NOW TRY