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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. 7.61 (page 678)

Simplify the complex rational expression by using the LCD:
$\frac{\frac{x^{2}}{x^{2} + 7 x + 10}}{\frac{4}{x + 2} + \frac{1}{x + 5}}$

Short Answer

Expert verified

The simplified expression is $\frac{x^{2}}{5 x + 22}$ .

Step by step solution

01

Step 1. Given Information.

We have:

$\frac{\frac{x^{2}}{x^{2} + 7 x + 10}}{\frac{4}{x + 2} + \frac{1}{x + 5}}$

02

Step 2. Find the LCD of all fractions in complex rational expression.

The least common divisor is $(x + 2) (x + 5) .$

Multiply the numerator and denominator by the LCD:

$= \frac{(x + 2) (x + 5) (\frac{x^{2}}{x^{2} + 7 x + 10})}{(x + 2) (x + 5) (\frac{4}{x + 2} + \frac{1}{x + 5})}$

03

Step 3. Simplify the expression.

Further, calculate:

$= \frac{(x + 2) (x + 5) (\frac{x^{2}}{x^{2} + 5 x + 2 x + 10})}{(x + 2) (x + 5) (\frac{4 (x + 5) + 1 (x + 2)}{(x + 2) (x + 5)})} = \frac{(x + 2) (x + 5) (\frac{x^{2}}{x (x + 5) + 2 (x + 5)})}{(x + 2) (x + 5) (\frac{4 (x + 5) + 1 (x + 2)}{(x + 2) (x + 5)})} = \frac{(x + 2) (x + 5) (\frac{x^{2}}{(x + 5) (x + 2)})}{(x + 2) (x + 5) (\frac{4 (x + 5) + 1 (x + 2)}{(x + 2) (x + 5)})} = \frac{x^{2}}{4 x + 20 + x + 2} = \frac{x^{2}}{5 x + 22}$

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