# How do you combine (x-2)/(x-6)-(x+2)/(6-x)?

Jul 22, 2016

$\frac{2 x}{x - 6}$

#### Explanation:

The trick here is to realize that you can write $6 - x$ as

$6 - x = - 1 \cdot \left(- 6 + x\right) = - 1 \cdot \left(x - 6\right)$

This means that the expression can be rewritten as

$\frac{x - 2}{x - 6} - \frac{x + 2}{- 1 \cdot \left(x - 6\right)} = \frac{x - 2}{x - 6} - \frac{x + 2}{- \left(x - 6\right)}$

This will be equivalent to

$\frac{x - 2}{x - 6} + \frac{x + 2}{x - 6}$

You now have two fractions that have the same denominator, which means that you can go ahead and combine their numerators

$\frac{x - 2}{x - 6} + \frac{x + 2}{x - 6} = \frac{x - \textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} + x + \textcolor{red}{\cancel{\textcolor{b l a c k}{2}}}}{x - 6} = \textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\frac{2 x}{x - 6}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

ALTERNATIVELY

You can also combine these two fractions by finding their common denominator, which in this case would be $\left(x - 6\right) \left(6 - x\right)$.

Multiply the first fraction by $1 = \frac{6 - x}{6 - x}$ and the second fraction by $1 = \frac{x - 6}{x - 6}$ to get

$\frac{x - 2}{x - 6} - \frac{x + 2}{6 - x} = \frac{x - 2}{x - 6} \cdot \frac{6 - x}{6 - x} - \frac{x + 2}{6 - x} \cdot \frac{6 - x}{6 - x}$

$= \frac{\left(x - 2\right) \left(6 - x\right)}{\left(x - 6\right) \left(6 - x\right)} - \frac{\left(x + 2\right) \left(6 - x\right)}{\left(x - 6\right) \left(6 - x\right)}$

$= \frac{\left(x - 2\right) \left(6 - x\right) - \left(x + 2\right) \left(x - 6\right)}{\left(x - 6\right) \left(6 - x\right)}$

Focus on the numerator first

$6 x - {x}^{2} - \textcolor{red}{\cancel{\textcolor{b l a c k}{12}}} + \textcolor{red}{\cancel{\textcolor{b l a c k}{2 x}}} - {x}^{2} + 6 x - \textcolor{red}{\cancel{\textcolor{b l a c k}{2 x}}} + \textcolor{red}{\cancel{\textcolor{b l a c k}{12}}} = - 2 {x}^{2} + 12 x$

This can be rewritten as

$- 2 {x}^{2} + 12 x = 12 x - 2 {x}^{2} = 2 x \cdot \left(6 - x\right)$

The expression will once again be equal to

$\frac{2 x \cdot \textcolor{red}{\cancel{\textcolor{b l a c k}{\left(6 - x\right)}}}}{\left(x - 6\right) \textcolor{red}{\cancel{\textcolor{b l a c k}{\left(6 - x\right)}}}} = \textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\frac{2 x}{x - 6}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

Keep in mind that you need to have $x \ne 6$.