Question

In the following exercise, find

(a) $R\left(x\right)=f\left(x\right)+g\left(x\right)$

(b) $R\left(x\right)=f\left(x\right)-g\left(x\right)$:

role="math" localid="1644749808906" $$\begin{gathered} f\left(x\right)=\frac{5}{x+7} \\ g\left(x\right)=\frac{10x}{x^2-49} \end{gathered}$$

Short answer

(a) $R\left(x\right)=\frac{5\left(3x-7\right)}{\left(x+7\right)\left(x-7\right)}$

(b)$R\left(x\right)=\frac{-5}{\left(x-7\right)}$

Step-by-step solution

Step 1. Given information.

The following information is given:

$$\begin{gathered} f\left(x\right)=\frac{5}{x+7} \\ g\left(x\right)=\frac{10x}{x^2-49} \end{gathered}$$

Step 2. Finding (a) Rx=fx+gx

$R\left(x\right)=f\left(x\right)+g\left(x\right)$

Substituting the values

role="math" localid="1644749830492" $R\left(x\right)=\frac{5}{x+7}+\frac{10x}{x^2-49}$

Factor the denominator:

role="math" localid="1644749841956" $R\left(x\right)=\frac{5}{x+7}+\frac{10x}{\left(x-7\right)\left(x+7\right)}$

Find out the LCD:

$\left(x-7\right)\left(x+7\right)$

Rewrite the expression with the denominator equal to the LCD:

$R\left(x\right)=\frac{5\left(x-7\right)}{\left(x+7\right)\left(x-7\right)}+\frac{10x}{\left(x-7\right)\left(x+7\right)}$

Simplify:

$$\begin{gathered} R\left(x\right)=\frac{5x-35+10x}{\left(x+7\right)\left(x-7\right)} \\ R\left(x\right)=\frac{15x-35}{\left(x+7\right)\left(x-7\right)} \\ R\left(x\right)=\frac{5\left(3x-7\right)}{\left(x+7\right)\left(x-7\right)} \end{gathered}$$

Step 3. Finding (b) Rx=fx-gx

$R\left(x\right)=f\left(x\right)-g\left(x\right)$

Substituting the values (after the denominator factored and made equal to LCD):

$R\left(x\right)=\frac{5\left(x-7\right)}{\left(x-7\right)\left(x+7\right)}-\frac{10x}{\left(x-7\right)\left(x+7\right)}$

Simplify:

$$\begin{gathered} R\left(x\right)=\frac{5x-35-10x}{\left(x-7\right)\left(x+7\right)} \\ R\left(x\right)=\frac{-5\left(x+7\right)}{\left(x-7\right)\left(x+7\right)} \end{gathered}$$

Cancelling common factors from the numerator and the denominator:

$R\left(x\right)=\frac{-5}{\left(x-7\right)}$