# How do you simplify (4x-9)/(2x-3)?

Jun 17, 2018

$\frac{4 x - 9}{2 x - 3} = \textcolor{b l u e}{2 - \frac{3}{2 x - 3}} \textcolor{w h i t e}{\text{xxxxx}}$...but it is not clear that this is a "simplification"

#### Explanation:

Since this was asked under "Excluded Values for Rational Expression" perhaps it should be noted that
$\textcolor{w h i t e}{\text{XXX}} \frac{4 x - 9}{2 x - 3}$ is not defined if $\left(2 x - 3\right) = 0 \rightarrow x = \frac{3}{2}$

Jun 17, 2018

$2 - \frac{3}{2 x - 3}$

#### Explanation:

Given: $\frac{4 x - 9}{2 x - 3}$

Starting the same process as long division (this one is short)

$\textcolor{w h i t e}{\text{ddddd.ddddd}} 4 x - 9$
$\textcolor{m a \ge n t a}{2} \left(2 x - 3\right) \to \underline{4 x - 6 \leftarrow \text{ Subtract}}$
$\textcolor{w h i t e}{\text{dddddddddddd")0color(magenta)(-3 larr" Remainder}}$

$\left(4 x - 9\right) \div \left(2 x - 3\right) = \textcolor{m a \ge n t a}{2 - \frac{3}{2 x - 3}}$