Question

In the following exercises, simplify.

$\frac{2-\frac{2}{a+3}}{\frac{1}{a+3}+\frac{a}{2}}$

Short answer

The simplified form of the given expression is$\frac{4}{a+1}.$

Step-by-step solution

Step 1. Given information.

The given expression is:

$\frac{2-\frac{2}{a+3}}{\frac{1}{a+3}+\frac{a}{2}}$

Step 2. Concept.

The following concept is required for simplification:

$\frac{A}{B}÷\frac{C}{D}=\frac{A}{B}·\frac{D}{C}$

Step 3. Rewrite the expression.

Simplify the numerator and denominator.

$$\begin{gathered} =\frac{\frac{2(a+3)-2}{a+3}}{\frac{2+a(a+3)}{2(a+3)}} \\ =\frac{\frac{2a+6-2}{a+3}}{\frac{2+a^2+3a)}{2(a+3)}} \\ =\frac{\frac{2a+4}{a+3}}{\frac{a^2+3a+2}{2(a+3)}} \\ =\frac{2a+4}{a+3}÷\frac{a^2+3a+2}{2(a+3)} \end{gathered}$$

Step 4. Simplify the expression.

Rewrite as the product of first times the reciprocal of the second.$$\begin{gathered} =\frac{2a+4}{a+3}·\frac{2(a+3)}{a^2+3a+2} \\ =\frac{2a+4}{a+3}·\frac{2(a+3)}{a^2+2a+a+2} \\ =\frac{2(a+2)}{a+3}·\frac{2(a+3)}{a(a+2)+1(a+2)} \\ =\frac{2(a+2)}{a+3}·\frac{2(a+3)}{(a+2)(a+1)} \\ =\frac{4}{a+1} \end{gathered}$$

Step 5. Conclusion.

The simplified form of the given expression is $\frac{4}{a+1}$.