Question

In the following exercises, simplify:

$\frac{{\displaystyle \frac{2}{x+5}}}{{\displaystyle \frac{3}{x-5}}+{\displaystyle \frac{1}{x^2-25}}}$.

Short answer

The simplified rational expression is,$\frac{2x-10}{3x+16}$.

Step-by-step solution

Step 1. Given Information.

The rational expression is,

$\frac{{\displaystyle \frac{2}{x+5}}}{{\displaystyle \frac{3}{x-5}}+{\displaystyle \frac{1}{x^2-25}}}$.

Step 2. Simplifying the rational expression.

$\frac{{\displaystyle \frac{2}{x+5}}}{{\displaystyle \frac{3}{x-5}}+{\displaystyle \frac{1}{x^2-25}}}$

The LCD of the expression is, $(x+5)(x-5)$,

$$\begin{gathered} =\frac{(x+5)(x-5){\displaystyle \frac{2}{x+5}}}{(x+5)(x-5)({\displaystyle \frac{3}{x-5}}+{\displaystyle \frac{1}{x^2-25}})} \\ =\frac{(x+5)(x-5)\left({\displaystyle \frac{2}{x+5}}\right)}{(x+5)(x-5)\left({\displaystyle \frac{3}{x-5}}\right)+(x+5)(x-5)\left({\displaystyle \frac{1}{(x+5)(x-5)}}\right)} \\ =\frac{2(x-5)}{3(x+5)+1} \\ =\frac{2x-10}{3x+15+1} \\ =\frac{2x-10}{3x+16} \end{gathered}$$