Question

In the following exercise, find

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

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

$$\begin{gathered} f\left(x\right)=\frac{-4x-24}{x^2+x-30} \\ g\left(x\right)=\frac{x+7}{5-x} \end{gathered}$$

Short answer

(a) $R\left(x\right)=\frac{-x-11}{x-5}$

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

Step-by-step solution

Step 1. Given information.

The following information is given:

$$\begin{gathered} f\left(x\right)=\frac{-4x-24}{x^2+x-30} \\ g\left(x\right)=\frac{x+7}{5-x} \end{gathered}$$

Step 2. Finding (a) Rx=fx+gx

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

Substituting the values

$R\left(x\right)=\frac{-4x-24}{x^2+x-30}+\frac{x+7}{5-x}$

Factorise the denominators:

$R\left(x\right)=\frac{-4x-24}{\left(x+6\right)\left(x-5\right)}+\frac{x+7}{5-x}$

Multiply the numerator and the denominator of the second term by $-1$and then simplify:

$$\begin{gathered} R\left(x\right)=\frac{-4x-24}{\left(x+6\right)\left(x-5\right)}+\frac{\left(-1\right)\left(x+7\right)}{\left(-1\right)\left(5-x\right)} \\ R\left(x\right)=\frac{-4\left(x+6\right)}{\left(x+6\right)\left(x-5\right)}-\frac{\left(x+7\right)}{(x-5)} \end{gathered}$$

Cancel the common factors from the numerator and the denominator:

$R\left(x\right)=\frac{-4}{\left(x-5\right)}-\frac{\left(x+7\right)}{(x-5)}$

Since common LCD is present, simplify:

$$\begin{gathered} R\left(x\right)=\frac{-4-x-7}{\left(x-5\right)} \\ R\left(x\right)=\frac{-x-11}{x-5} \end{gathered}$$

Step 3. Find out (b) Rx=f(x)-g(x)

By factorising the terms,

$R\left(x\right)=\frac{-4\left(x+6\right)}{\left(x+6\right)\left(x-5\right)}-\frac{x+7}{5-x}$

Cancelling $x+6$from the first term and multiplying both the numerator and the denominator of the second term by$-1$and then simplifying:

localid="1644743911735" $$\begin{gathered} R\left(x\right)=\frac{-4}{\left(x-5\right)}-\frac{\left(-1\right)\left(x+7\right)}{\left(-1\right)\left(5-x\right)} \\ R\left(x\right)=\frac{-4}{\left(x-5\right)}+\frac{\left(x+7\right)}{\left(x-5\right)} \end{gathered}$$

Simplify:

$$\begin{gathered} R\left(x\right)=\frac{-4+x+7}{\left(x-5\right)} \\ R\left(x\right)=\frac{x+3}{x-5} \end{gathered}$$