Question

In the following exercises, simplify.

$\frac{7+{\displaystyle \frac{2}{q-2}}}{{\displaystyle \frac{1}{q+2}}}$

Short answer

The simplified expression is$\frac{7q^2+2q-24}{q-2}$

Step-by-step solution

Step1. Given Information

The given expression is$\frac{7+{\displaystyle \frac{2}{q-2}}}{{\displaystyle \frac{1}{q+2}}}$

Step 2. Simplify the expression 

$$\begin{gathered} \frac{7+{\displaystyle \frac{2}{q-2}}}{{\displaystyle \frac{1}{q+2}}}=\frac{\left(\cancel{q+2}\right)\left(\cancel{q-2}\right)\left(7+{\displaystyle \frac{2}{\left(\cancel{q-2}\right)}}\right)}{\left(\cancel{q+2}\right)\left(\cancel{q-2}\right)\left({\displaystyle \frac{1}{\cancel{q+2}}}\right)} \\ =\frac{7\left(q+2\right)\left(q-2\right)+2\left(q+2\right)}{q-2} \\ =\frac{7q^2-28+2q+4}{q-2} \\ =\frac{7q^2+2q-24}{q-2} \end{gathered}$$

The simplified expression is$\frac{7q^2+2q-24}{q-2}$

Step 3. The solution

The simplified expression is$\frac{7q^2+2q-24}{q-2}$