Question

In the following exercise find $R\left(x\right)=f\left(x\right)-g\left(x\right)$where $f\left(x\right)$and$g\left(x\right)$are givenrole="math" localid="1646710113564" $f\left(x\right)=\frac{7}{x+6},g\left(x\right)=\frac{14x}{x^2-36}$

Short answer

The solution is$-\frac{7}{x-6}$

Step-by-step solution

Step 1. Given information

$f\left(x\right)=\frac{7}{x+6},g\left(x\right)=\frac{14x}{x^2-36}$

Step 2. Find Rx=fx-gx

The LCD of $x+6,x^2-36:\left(x+6\right)\left(x-6\right)$

Rewrite each rational expression as an equivalent rational expression with LCD. localid="1663033675009" $$\begin{gathered} R\left(x\right)=f\left(x\right)-g\left(x\right) \\ R\left(x\right)=\frac{7\left(x-6\right)}{\left(x+6\right)\left(x-6\right)}-\frac{14x}{\left(x+6\right)\left(x-6\right)} \\ R\left(x\right)=\frac{7x-42-14x}{\left(x+6\right)\left(x-6\right)} \\ R\left(x\right)=\frac{-7\left(x+6\right)}{\left(x+6\right)\left(x-6\right)} \\ R\left(x\right)=-\frac{7}{\left(x-6\right)} \end{gathered}$$