Question

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

Answer 1

`x=-2` and `x=10`

Explanation:

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

`abs(x-4)-6<6-6`

This becomes:

`abs(x-4)-6<0`

Let's focus on what `abs(x-4)` 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`