# How do you graph and solve  |x-4|<9?

Mar 3, 2017

$- 5 < x < 13$

#### Explanation:

When working with absolute value inequalities, we need to remember that the absolute value function will return a positive value regardless of what lies within. For instance, if we have

$x = 3$, $x$ can be 3 or it can be $- 3$. And so in our question we need to address this:

$\left\mid x - 4 \right\mid < 9 \implies x - 4 < \pm 9 \implies - 9 < x - 4 < 9$

We can now solve this by adding 4 to all sides:

$- 9 < x - 4 < 9$

$- 9 \textcolor{red}{+ 4} < x - 4 \textcolor{red}{+ 4} < 9 \textcolor{red}{+ 4}$

$- 5 < x < 13$

We can graph this on a number line by placing hollow circles around $- 5$ and 13 (to indicate that all the points up to but not including $- 5$ and 13 are part of the solution) and then drawing a line connecting the two.