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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 429 (page 737)

In the following exercise, simplify
$\frac{x - \frac{3 x}{x + 5}}{\frac{1}{x + 5} + \frac{1}{x - 5}}$

Short Answer

Expert verified

The solution is $\frac{(x + 2) (x - 5)}{2}$

Step by step solution

01

Step 1. Given Information

$\frac{x - \frac{3 x}{x + 5}}{\frac{1}{x + 5} + \frac{1}{x - 5}}$

02

Step 2. Simplify.

Simplify the numerator and denominator.

Find the LCD and add the fractions in the numerator.

Find the LCD and subtract the fractions in the denominator.

$= \frac{\frac{x (x + 5) - 3 x}{(x + 5)}}{\frac{(x - 5) + (x + 5)}{(x + 5) (x - 5)}} = \frac{\frac{x^{2} + 2 x}{(x + 5)}}{\frac{2 x}{(x + 5) (x - 5)}}$

Rewrite the complex rational expression as a division problem.

$= \frac{x^{2} + 2 x}{(x + 5)} 梅 \frac{(2 x)}{(x + 5) (x - 5)}$

Multiply the first by the reciprocal of the second.

$= \frac{x^{2} + 2 x}{(x + 5)} 脳 \frac{(x + 5) (x - 5)}{(2 x)}$

Remove common factors.

$\frac{(x^{2} + 2 x) (x - 5)}{2 x}$

Simplify

$\frac{(x + 2) (x - 5)}{2}$

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Most popular questions from this chapter

Simplify the complex rational expression by writing it as division:

$\frac{\frac{1}{x} + \frac{1}{y}}{\frac{1}{x} - \frac{1}{y}}$

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