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

Question

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

Eva Abramson · Accepted Answer

#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#

Julia Adams · Answer

undefined To graph and solve the inequality $|x-4| < 6$, first, set up two inequalities representing the positive and negative cases of the absolute value:

1. $x - 4 < 6$
2. $-(x - 4) < 6$

Solve each inequality separately:

1. $x - 4 < 6$
   Add 4 to both sides:
   $x < 10$

2. $-(x - 4) < 6$
   Distribute the negative sign:
   $-x + 4 < 6$
   Subtract 4 from both sides:
   $-x < 2$
   Multiply both sides by -1 (remember to flip the inequality sign):
   $x > -2$

So, the solution to the inequality is $-2 < x < 10$.

To graph the solution, draw a number line and mark -2 and 10 with open circles, indicating that they are not included in the solution. Then shade the region between -2 and 10 to represent all values of $x$ that satisfy the inequality.

How do you graph and solve #abs(x-4) &lt;6 #?

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