Question

How to easily understand the concept of absolute value inequalities and absolute value equations? See the comment below for the solution.

Answer 1

Absolute value equations and inequalities confuse everyone. Though, the concept is very simple. If you read this post with all your senses, you'll master this.

Here, let's take an example:
\[\large {|x| < 5} \]

Now, replace the |x| with "distance from zero" and read it out aloud. You read:
"Distance from zero is less than 5". So, imagine a number line. 5 away from zero can either be 5 or -5. We know that distance from zero is less than 5.
We can say this.
\[\large {x > -5} \]
and
\[\large {x < 5} \]

So, simplifying this statement, we can say that:
\[\large {-5 < x < 5} \]

Another example.

\[\large {|x| > 23 }\]. Now replace '|x|' with distance from zero, and read it out aloud:-
"Distance from zero is more than 23"

We can say this.
\[\large {x > 23} or {x < -23} \]

If you know how to solve for x in inequalities, then you can easily solve |2x| or |x - 1|.

Same is with equations.

\[\large |x| = 9 \]. Replace '|x|' with 'Distance from zero' and read it:
"Distance from zero is equal to 9."

So, x = 9 or x = -9