Question

Simplify the complex rational expression by using the LCD:

$\frac{{\displaystyle \frac{3}{x+2}}}{{\displaystyle \frac{5}{x-2}}-{\displaystyle \frac{3}{x^2-4}}}$

Short answer

$\frac{3x-6}{5x+7}$

Step-by-step solution

Step 1. Given information.

The following complex rational expression is given:

$\frac{{\displaystyle \frac{3}{x+2}}}{{\displaystyle \frac{5}{x-2}-\frac{3}{x^2-4}}}$

Step 2. Simplification of the complex rational expression.

The LCD of all the fractions the numerator and the denominator is $x^2-4=\left(x+2\right)\left(x-2\right)$.

Multiplying both the numerator and the denominator by LCD:

$\frac{{\displaystyle \left(x^2-4\right).\frac{3}{x+2}}}{{\displaystyle \left(x^2-4\right).\left(\frac{5}{x-2}-\frac{3}{x^2-4}\right)}}$

Distributing and simplifying:

$$\begin{gathered} \frac{{\displaystyle \left(x+2\right)\left(x-2\right).\frac{3}{x+2}}}{{\displaystyle \left(x+2\right)\left(x-2\right).\frac{5}{x-2}-\left(x^2-4\right).\frac{3}{x^2-4}}} \\ \frac{3\left(x-2\right)}{5\left(x+2\right)-3} \end{gathered}$$

Simplifying:

$\frac{3x-6}{5x+7}$