Question

simplify.\\(
\frac{1 - \frac{3}{x + 6}}{x + \frac{9}{x + 6}}
\\)

Explanation:

Step1: Simplify numerator

Rewrite 1 as $\frac{x+6}{x+6}$, then subtract:
$$1 - \frac{3}{x+6} = \frac{x+6}{x+6} - \frac{3}{x+6} = \frac{x+6-3}{x+6} = \frac{x+3}{x+6}$$

Step2: Simplify denominator

Rewrite $x$ as $\frac{x(x+6)}{x+6}$, then add:
$$x + \frac{9}{x+6} = \frac{x(x+6)}{x+6} + \frac{9}{x+6} = \frac{x^2+6x+9}{x+6} = \frac{(x+3)^2}{x+6}$$

Step3: Divide numerator by denominator

Multiply by reciprocal of denominator:
$$\frac{\frac{x+3}{x+6}}{\frac{(x+3)^2}{x+6}} = \frac{x+3}{x+6} \times \frac{x+6}{(x+3)^2}$$

Step4: Cancel common factors

Cancel $x+6$ and one $x+3$ term:
$$\frac{1}{x+3}$$

Answer:

$\frac{1}{x+3}$