# How do you combine 4/(a-5) - 1/(5-a)?

Mar 2, 2018

The combined fraction is $\frac{5}{a - 5}$.

#### Explanation:

Rewrite the subtraction as an addition of a negative number:

$\textcolor{w h i t e}{=} \frac{4}{a - 5} - \frac{1}{5 - a}$

$= \frac{4}{a - 5} + \left(- \frac{1}{5 - a}\right)$

Next, instead of putting that negative sign next to the $1$ on the numerator, put it in the denominator, like this:

$\textcolor{w h i t e}{=} \frac{4}{a - 5} + \left(- \frac{1}{5 - a}\right)$

$= \frac{4}{a - 5} + \frac{1}{- \left(5 - a\right)}$

Now, just simplify:

$\textcolor{w h i t e}{=} \frac{4}{a - 5} + \frac{1}{- \left(5 - a\right)}$

$= \frac{4}{a - 5} + \frac{1}{- 5 + a}$

$= \frac{4}{a - 5} + \frac{1}{a - 5}$

$= \frac{4 + 1}{a - 5}$

$= \frac{5}{a - 5}$