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

May 14, 2017

$\frac{x - 3}{x + 2} = 1 - \frac{5}{x + 2}$

#### Explanation:

As a rational expression,

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

However, it is possible to separate it out into the sum of a trivial polynomial and a partial fraction with constant numerator:

$\frac{x - 3}{x + 2} = \frac{\left(x + 2\right) - 5}{x + 2} = 1 - \frac{5}{x + 2}$

One advantage of this form is that $x$ only occurs once, so it is easy to invert as a function:

Let:

$y = 1 - \frac{5}{x + 2}$

Add $\frac{5}{x + 2} - y$ to both sides to get:

$\frac{5}{x + 2} = 1 - y$

Take the reciprocal of both sides to get:

$\frac{x + 2}{5} = \frac{1}{1 - y}$

Multiply both sides by $5$ to get:

$x + 2 = \frac{5}{1 - y}$

Subtract $2$ from both sides to get:

$x = \frac{5}{1 - y} - 2$

So if:

$f \left(x\right) = \frac{x - 3}{x + 2} = 1 - \frac{5}{x + 2}$

then:

${f}^{- 1} \left(y\right) = \frac{5}{1 - y} - 2$