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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. 7.62 (page 678)

Simplify the complex rational expression by using the LCD:
$\frac{\frac{4 y}{y + 5} + \frac{2}{y + 6}}{\frac{3 y}{y^{2} + 11 y + 30}}$

Short Answer

Expert verified

The simplified expression is $\frac{(2 y^{2} + 13 y + 5)}{3 y}$ .

Step by step solution

01

Step 1. Given Information.

We have:

$\frac{\frac{4 y}{y + 5} + \frac{2}{y + 6}}{\frac{3 y}{y^{2} + 11 y + 30}}$

02

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

The least common divisor is $(y + 5) (y + 6)$ .

Multiply the numerator and denominator by the LCD:

$= \frac{(y + 5) (y + 6) (\frac{4 y}{y + 5} + \frac{2}{y + 6})}{(y + 5) (y + 6) (\frac{3 y}{y^{2} + 11 y + 30})}$

03

Step 3. Simplify the expression.

Further, calculate:

$= \frac{(y + 5) (y + 6) (\frac{4 y}{(y + 5)} + \frac{2}{(y + 6)})}{(y + 5) (y + 6) (\frac{3 y}{y^{2} + 5 y + 6 y + 30})} = \frac{(y + 5) (y + 6) (\frac{4 y (y + 6) + 2 (y + 5)}{(y + 5) (y + 6)})}{(y + 5) (y + 6) (\frac{3 y}{y (y + 5) + 6 (y + 5)})} = \frac{(y + 5) (y + 6) (\frac{4 y^{2} + 24 y + 4 y + 10}{(y + 5) (y + 6)})}{(y + 5) (y + 6) (\frac{3 y}{(y + 5) (y + 6)})} = \frac{4 y^{2} + 26 y + 10}{3 y} = \frac{2 (2 y^{2} + 13 y + 5)}{3 y}$

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