# How do you simplify \frac { \frac { x } { 10} - \frac { 5} { x } } { \frac { 1} { 5} + \frac { 1} { x } }?

Aug 27, 2017

Make all sub-fractions have the same denominator; cancel this off. Factor and reduce any remaining terms common to both sides of the overall fraction.

#### Explanation:

Step 1: Give all terms a common denominator (in this case, $10 x$).

$\frac{\frac{x}{10} - \frac{5}{x}}{\frac{1}{5} + \frac{1}{x}} = \frac{\frac{x}{10} \left(\frac{x}{x}\right) - \frac{5}{x} \left(\frac{10}{10}\right)}{\frac{1}{5} \left(\frac{2 x}{2 x}\right) + \frac{1}{x} \left(\frac{10}{10}\right)}$

$\textcolor{w h i t e}{\frac{\frac{x}{10} - \frac{5}{x}}{\frac{1}{5} + \frac{1}{x}}} = \frac{{x}^{2} / \left(10 x\right) - \frac{50}{10 x}}{\frac{2 x}{10 x} + \frac{10}{10 x}}$

Step 2: Cancel off this common denominator (with the condition that its value cannot be zero).

$\textcolor{w h i t e}{\frac{\frac{x}{10} - \frac{5}{x}}{\frac{1}{5} + \frac{1}{x}}} = \frac{{x}^{2} - 50}{2 x + 10} \text{ }$ (if $x \ne 0$)

Step 3: Factor if requested/possible.

$\textcolor{w h i t e}{\frac{\frac{x}{10} - \frac{5}{x}}{\frac{1}{5} + \frac{1}{x}}} = \frac{{x}^{2} - 50}{2 \left(x + 5\right)}$

Step 4: Cancel any common factors (in this case, there are none).

$\textcolor{w h i t e}{\frac{\frac{x}{10} - \frac{5}{x}}{\frac{1}{5} + \frac{1}{x}}} = \frac{{x}^{2} - 50}{2 \left(x + 5\right)}$