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

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Answer 1 · 2017-05-31T01:35:34

#x=-2# and #x=10#

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#