How do you simplify (9 - x^(-2))/(3 + x^(-1))?

Oct 4, 2015

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

Explanation:

Since
$\textcolor{w h i t e}{\text{XXX} \textcolor{red}{} 9 - \frac{1}{x} ^ \left(- 2\right)}$
$\textcolor{w h i t e}{\text{XXXXXX}} = \textcolor{red}{9 - \frac{1}{{x}^{2}}}$

$\textcolor{w h i t e}{\text{XXXXXX}} = \textcolor{red}{\frac{9 {x}^{2} - 1}{{x}^{2}}}$
and
$\textcolor{w h i t e}{\text{XXX}} \textcolor{b l u e}{3 + {x}^{- 1}}$
$\textcolor{w h i t e}{\text{XXXXXX}} = \textcolor{b l u e}{3 + \frac{1}{x}}$

$\textcolor{w h i t e}{\text{XXXXXX}} = \textcolor{b l u e}{\frac{3 x + 1}{x}}$

$\frac{\textcolor{red}{\left(9 - \frac{1}{{x}^{2}}\right)}}{\textcolor{b l u e}{\left(3 + {x}^{- 1}\right)}}$

$\textcolor{w h i t e}{\text{XXX}} = \frac{\textcolor{red}{\frac{9 {x}^{2} - 1}{{x}^{2}}}}{\textcolor{b l u e}{\frac{3 x + 1}{x}}}$

$\textcolor{w h i t e}{\text{XXX}} = \frac{9 {x}^{2} - 1}{{x}^{2}} \cdot \frac{x}{3 x + 1}$

$\textcolor{w h i t e}{\text{XXX}} = \frac{\left(3 x + 1\right) \left(3 x - 1\right)}{x \cdot x} \cdot \frac{x}{3 x + 1}$

$\textcolor{w h i t e}{\text{XXX}} = \frac{\cancel{\left(3 x + 1\right)} \left(3 x - 1\right)}{\cancel{x} \cdot x} \cdot \frac{\cancel{x}}{\cancel{\left(3 x + 1\right)}}$

$\textcolor{w h i t e}{\text{XXX}} = \frac{3 x - 1}{x}$

$\textcolor{w h i t e}{\text{XXX}} = 3 - \frac{1}{x}$