Question

In the following exercises, simplify.

$\frac{{\displaystyle \frac{5}{c+2}}-{\displaystyle \frac{3}{c+7}}}{{\displaystyle \frac{5c}{c^2+9c+14}}}$

Short answer

The simplified expression is$\frac{2c+29}{5c}$

Step-by-step solution

Step 1. Given information

The given expression is

$\frac{{\displaystyle \frac{5}{c+2}-\frac{3}{c+7}}}{{\displaystyle \frac{5c}{c^2+9c+14}}}$

Step 2. Simplify the expression

$$\begin{gathered} \frac{{\displaystyle \frac{5}{c+2}-\frac{3}{c+7}}}{{\displaystyle \frac{5c}{c^2+9c+14}}}=\frac{\left(\cancel{c+2}\right)\left(\cancel{c+7}\right)\left({\displaystyle \frac{5}{\left(\cancel{c+2}\right)}}-{\displaystyle \frac{3}{\left(\cancel{c+7}\right)}}\right)}{\left(\cancel{c+2}\right)\left(\cancel{c+7}\right)\left({\displaystyle \frac{5c}{\left(\cancel{c+3}\right)\left(\cancel{c+7}\right)}}\right)} \\ =\frac{5\left(c+7\right)-3\left(c+2\right)}{5c} \\ =\frac{2c+29}{5c} \end{gathered}$$

The simplified expression is$\frac{2c+29}{5c}$

Step 3. The solution 

The simplified solution is$\frac{2c+29}{5c}$