Question

In the following exercise, simplify

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

Short answer

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

Step-by-step solution

Step 1. Given Information

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

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.

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

Rewrite the complex rational expression as a division problem.

$=\frac{x^2+2x}{\left(x+5\right)}÷\frac{\left(2x\right)}{\left(x+5\right)\left(x-5\right)}$

Multiply the first by the reciprocal of the second.

$=\frac{x^2+2x}{\left(x+5\right)}\times \frac{\left(x+5\right)\left(x-5\right)}{\left(2x\right)}$

Remove common factors.

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

Simplify

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