# How do you solve and graph abs(m+19)<=1?

Apr 19, 2017

Normally, when we want to solve for $x$, we move everything besides $x$ to the other side. But what is the inverse of abs(color(white)(x+7)?

Well, the absolute value bars make the value inside of them positive. So, $\left\mid 2 \right\mid$ and $\left\mid - 2 \right\mid$ both equal $2$.

Based on that fact, $\left\mid m + 19 \right\mid$ could be based on two situations: $\left(m + 19\right)$ or $- 1 \left(m + 19\right)$. We need to solve for both of these situations:

Situation 1
$m + 19 \le 1$
subtract $19$ on both sides
$m \le - 18$

Situation 2
$- 1 \left(m + 19\right) \le 1$
divide by $- 1$ on both sides
$m + 19 = - 1$

Notice, the sign ($\le$) changed. Whenever we divide by a negative number, the sign switches

subtract by $19$ on both sides
$m \ge - 1 - 19$
$m \ge - 20$

So, our two values are $m \le - 18$ and $m \ge - 20$. These are the $x -$intercepts

graph{abs(x+19)-1}