# How do you simplify 2/(x-2) - 3/(x-1)?

Feb 15, 2016

Use multiplication by $1$ to make the denominators the same to find

$\frac{2}{x - 2} - \frac{3}{x - 1} = \frac{- x + 4}{\left(x - 1\right) \left(x - 2\right)}$

#### Explanation:

$\frac{2}{x - 2} - \frac{3}{x - 1} = \frac{2}{x - 2} \cdot \textcolor{red}{1} - \frac{3}{x - 1} \cdot \textcolor{red}{1}$

$= \frac{2}{x - 2} \cdot \textcolor{red}{\frac{x - 1}{x - 1}} - \frac{3}{x - 1} \cdot \textcolor{red}{\frac{x - 2}{x - 2}}$

$= \frac{2 \left(x - 1\right)}{\left(x - 1\right) \left(x - 2\right)} - \frac{3 \left(x - 2\right)}{\left(x - 1\right) \left(x - 2\right)}$

$= \frac{2 \left(x - 1\right) - 3 \left(x - 2\right)}{\left(x - 1\right) \left(x - 2\right)}$

$= \frac{2 x - 2 - 3 x + 6}{\left(x - 1\right) \left(x - 2\right)}$

$= \frac{- x + 4}{\left(x - 1\right) \left(x - 2\right)}$