# How do you write m(x) = |x+2| - 7 as a piecewise function?

Apr 28, 2017

Use the fact that $\left\mid u \right\mid = \left\{\begin{matrix}u & \text{if" & u >= 0 \\ -u & "if} & u < 0\end{matrix}\right.$

#### Explanation:

So

$m \left(x\right) = \left\{\begin{matrix}\left(x + 2\right) - 7 & \text{if" & (x+2) >= 0 \\ -(x+2)-7 & "if} & \left(x + 2\right) < 0\end{matrix}\right.$

We can simplify the expressions

$\left(x + 2\right) - 7 = x - 5$ and $- \left(x + 2\right) - 7 = - x - 9$

and we can solve the inequalities

$\left(x + 2\right) \ge 0$ is equivalent to $x \ge - 2$ and
$\left(x + 2\right) < 0$ is equivalent to $x < - 2$

So we can write the function:

$m \left(x\right) = \left\{\begin{matrix}x - 5 & \text{if" & x >= -2 \\ -x-9 & "if} & x < - 2\end{matrix}\right.$.