Question

Simplify the complex rational expression by using the LCD:

$\frac{{\displaystyle \frac{2}{x-7}}-{\displaystyle \frac{1}{x+7}}}{{\displaystyle \frac{6}{x+7}}-{\displaystyle \frac{1}{x^2-49}}}$

Short answer

$\frac{x+21}{6x-43}$

Step-by-step solution

Step 1. Given information.

The following complex rational expression is given:

$\frac{{\displaystyle \frac{2}{x-7}-\frac{1}{x+7}}}{{\displaystyle \frac{6}{x+7}-\frac{1}{x^2-49}}}$

Step 2. Simplification of the complex rational expression.

The LCD of the numerator and the denominator fractions is $x^2-49=\left(x+7\right)\left(x-7\right)$.

Multiplying both the numerator and the denominator by LCD and distributing:

$$\begin{gathered} \frac{{\displaystyle \left(x^2-49\right).\left(\frac{2}{x-7}-\frac{1}{x+7}\right)}}{{\displaystyle \left(x^2-49\right).\left(\frac{6}{x+7}-\frac{1}{x^2-49}\right)}} \\ \frac{{\displaystyle \left(x+7\right)\left(x-7\right).\frac{2}{x-7}-\left(x+7\right)\left(x-7\right).\frac{1}{x+7}}}{{\displaystyle \left(x+7\right)\left(x-7\right).\frac{6}{x+7}-\left(x^2-49\right).\frac{1}{x^2-49}}} \end{gathered}$$

Simplifying:

$\frac{{\displaystyle 2\left(x+7\right)-\left(x-7\right)}}{{\displaystyle 6\left(x-7\right)-1}}$

Distributing and simplifying:

$$\begin{gathered} \frac{2x+14-x+7}{6x-42-1} \\ \frac{x+21}{6x-43} \end{gathered}$$