# How do you graph and solve abs(x-4) <6 ?

May 31, 2017

$x = - 2$ and $x = 10$

#### Explanation:

Rewrite the problem so that everything is less than $0$ by subtracting $6$ on both sides:

$\left\mid x - 4 \right\mid - 6 < 6 - 6$

This becomes:

$\left\mid x - 4 \right\mid - 6 < 0$

Let's focus on what $\left\mid x - 4 \right\mid$ looks like. The absolute value function looks like this:

graph{|x| [-10, 10, -5, 5]}

Notice how the graph is always above $y = 0$. That's because of the absolute value.

Now the $- 4$ makes it so our graph will move to the right 4 units. Our graph will, therefore, look like:

graph{|x-4| [-10, 10, -5, 5]}

Now the $- 6$ will make it so this graph goes down $6$ units. Our graph will, therefore, look like:

graph{|x-4|-6 [-14.24, 14.24, -7.12, 7.12]}

We can't forget about the $<$ symbol. Simply shade under the graph. Make sure the lines of the graph are dashed to signifiy that it's not equal to.:

graph{y<|x-4|-6 [-22.81, 22.8, -11.4, 11.41]}

Now that we have successfully graphed this, we can solve it by finding the zeroes. This happens at $x = - 2$ and $x = 10$