Question

Simplify:$\frac{\frac{2}{x^2-1}}{\frac{3}{x+1}}$

Short answer

The simplified form of the given expression is $\frac{2}{3(x-1)}$.

Step-by-step solution

Step 1. Given information.

The given expression is:

$\frac{\frac{2}{x^2-1}}{\frac{3}{x+1}}$

Step 2. Concepts.

The following concepts are required for simplification:

$\begin{array}{rcl}\frac{A}{B}÷\frac{C}{D}& =& \frac{A}{B}·\frac{D}{C}\\ a^2-b^2& =& (a-b)(a+b)\end{array}$

Step 3. Rewrite the expression and then simplify.

Rewrite the complex fraction as division.

$\frac{2}{x^2-1}÷\frac{3}{x+1}$

Rewrite as the product of first times the reciprocal of the second.

$$\begin{gathered} =\frac{2}{x^2-1}·\frac{x+1}{3} \\ =\frac{2}{(x-1)(x+1)}·\frac{x+1}{3} \\ =\frac{2(x+1)}{3(x-1)(x+1)} \\ =\frac{2}{3(x-1)} \end{gathered}$$

Step 4. Conclusion.

The simplified form of the given expression is $\frac{2}{3(x-1)}$.