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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 7: Q7.54TI. (page 674)

Simplify the complex rational expression by writing it as division:
$\frac{1 - \frac{3}{c + 4}}{\frac{1}{c + 4} + \frac{c}{3}}$

Short Answer

Expert verified

$\frac{3}{c + 3}$

Step by step solution

01

Step 1. Given information.

The following complex rational expression is given:

$\frac{1 - \frac{3}{c + 4}}{\frac{1}{c + 4} + \frac{c}{3}}$

02

Step 2. Simplification of the complex rational expression.

Find common denominators and simplify the numerator and the denominator:

$\frac{\frac{1 (c + 4)}{(c + 4)} - \frac{3}{c + 4}}{\frac{1.3}{3 (c + 4)} + \frac{c (c + 4)}{3 (c + 4)}} \frac{\frac{c + 4 - 3}{(c + 4)}}{\frac{3 + c (c + 4)}{3 (c + 4)}} \frac{\frac{(c + 1)}{(c + 4)}}{\frac{3 + c^{2} + 4 c}{3 (c + 4)}} \frac{\frac{(c + 1)}{(c + 4)}}{\frac{c^{2} + 4 c + 3}{3 (c + 4)}}$

Rewrite as a division problem:

$\frac{(c + 1)}{(c + 4)} 梅 \frac{c^{2} + 4 c + 3}{3 (c + 4)}$

Rewrite as a product of the first with the reciprocal of the second:

$\frac{(c + 1)}{(c + 4)} . \frac{3 (c + 4)}{c^{2} + 4 c + 3}$

Factor the numerator and the denominator terms (where applicable), cancel the common factors and simplify:

$\frac{(c + 1)}{(c + 4)} . \frac{3 (c + 4)}{(c + 3) (c + 1)} \frac{3}{c + 3}$

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