Question

Simplify the complex rational expression by writing it as division:

$\frac{{\displaystyle \frac{1}{x^2-7x+12}}}{{\displaystyle \frac{2}{x-4}}}$

Short answer

The simplified answer is$\frac{1}{2\left(x-3\right)}$

Step-by-step solution

Step 1. Given information.

The following rational expression is given:

$\frac{{\displaystyle \frac{1}{x^2-7x+12}}}{{\displaystyle \frac{2}{x-4}}}$

Step 2. Simplify the rational expression by division.

Rewrite the complex fraction as division:

$\frac{1}{x^2-7x+12}÷\frac{2}{x-4}$

Rewrite as the product of the first and the reciprocal of the second:

$\frac{1}{x^2-7x+12}.\frac{x-4}{2}$

Now,

$$\begin{gathered} x^2-7x+12=x^2-4x-3x+12 \\ {x}^2-7x+12=x\left(x-4\right)-3\left(x-4\right) \\ {x}^2-7x+12=\left(x-4\right)\left(x-3\right) \end{gathered}$$

Factor:

$\frac{1}{\left(x-3\right)\left(x-4\right)}.\frac{x-4}{2}$

Multiply:

$\frac{\left(x-4\right)}{2\left(x-3\right)\left(x-4\right)}$

Remove common factors from the numerator and the denominator:

$\frac{1}{2\left(x-3\right)}$