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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 7: Q. 163 (page 680)

In the following exercises, simplify each complex rational expression by writing it as division.
$\frac{x - \frac{2 x}{x + 3}}{\frac{1}{x + 3} + \frac{1}{x - 3}}$

Short Answer

Expert verified

The simplified expression is $\frac{(x + 1) (x - 3)}{2}$ .

Step by step solution

01

Step 1. Given Information.

We have:

$\frac{x - \frac{2 x}{x + 3}}{\frac{1}{x + 3} + \frac{1}{x - 3}}$

02

Step 2. Simplify the numerator:

The numerator is $x - \frac{2 x}{x + 3}$ .

The least common divisor of $1, x + 3$ is $x + 3$ .

Adjust the fractions based on the LCM:

$= \frac{x (x + 3) - 2 x}{x + 3} = \frac{x^{2} + x}{x + 3}$

03

Step 3. Simplify the denominator.

The denominator is $\frac{1}{x + 3} + \frac{1}{x - 3}$ .

The least common divisor is $(x + 3) (x - 3)$ .

Adjust the fractions based on the LCM:

$= \frac{x - 3}{(x + 3) (x - 3)} + \frac{x + 3}{(x + 3) (x - 3)} = \frac{x - 3 + x + 3}{(x + 3) (x - 3)} = \frac{2 x}{(x + 3) (x - 3)}$

04

Step 4. Substitute the simplified numerator and denominator into the original expression.

We have:

$\frac{x - \frac{2 x}{x + 3}}{\frac{1}{x + 3} + \frac{1}{x - 3}} = \frac{\frac{x^{2} + x}{x + 3}}{\frac{2 x}{(x + 3) (x - 3)}}$

Apply fraction rule: $\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a 路鈥� d}{b 路鈥� c}$

$= \frac{(x^{2} + x) (x + 3) (x - 3)}{(x + 3) 路鈥� 2 x} = \frac{(x + 1) (x - 3)}{2}$

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