The absolute value inequality means x is between -6 and +6:
==> ==> (-6,6)
It is helpful to solve absolute value inequalities by interpreting to mean "the distance between x and a is less than b".
Your example is then
==> ==> the distance between x and 0 is less than 6.
That means x can be up to 6 units to the left of 0 or up to 6 units to the right of 0; and that gives the answer (-6,6).
Let's look at another slightly more complicated example using a formal algebraic solution and a solution using the interpretation of absolute value as the distance between points.
Solve
(1) Algebraically....
"x-4 is between -3 and +3":
Solution: (1,7)
(2) Using distances....
"the distance between x and 4 is less than 3"
3 to the left of 4 is 1; 3 to the right is 7
Solution: (1,7)
The solutions by both methods are relatively simple. But for more complicated absolute value problems, using the interpretation of absolute value as the distance between points is usually easier.
