# How do you simplify (4x-1)/(x^2-4) - (3(x-1))/(x-2)?

Jul 29, 2017

$= \frac{- 3 {x}^{2} + x + 5}{\left(x + 2\right) \left(x - 2\right)}$

#### Explanation:

Before you can subtract fractions you must have a common denominator. In algebraic fractions you will often have to factorise first.

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

$= \textcolor{b l u e}{\frac{4 x - 1}{\left(x + 2\right) \left(x - 2\right)}} - \frac{3 \left(x - 1\right)}{x - 2} \text{ } \leftarrow$LCD = $\left(x + 2\right) \left(x - 2\right)$
$\textcolor{w h i t e}{\times \times \times} \downarrow \textcolor{w h i t e}{\times \times \times \times \times} \downarrow$
$\textcolor{b l u e}{\text{stays the same}} \textcolor{w h i t e}{\times \times \setminus x} \times \textcolor{red}{\frac{x + 2}{x + 2}}$

$= \frac{\textcolor{b l u e}{4 x - 1} \textcolor{red}{- 3 \left(x - 1\right) \left(x + 2\right)}}{\left(x + 2\right) \left(x - 2\right)}$

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

$= \frac{4 x - 1 - \textcolor{red}{3 {x}^{2} - 3 x + 6}}{\left(x + 2\right) \left(x - 2\right)}$

$= \frac{- 3 {x}^{2} + x + 5}{\left(x + 2\right) \left(x - 2\right)}$