# Solving absolute value inequality

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For which values of x does the following inequality hold true:

(|x|+2)/(|x|-3) < 4

The first thing we want to do is eliminate the fraction on the left-hand side (LHS) of the equation. We do so by multiplying both side by |x|-3, leading to:

|x|+2 < 4(|x|-3)

We can simplify the right-hand side (RHS):

|x| + 2 < 4|x| - 12

We now want to group like terms together. As such, we will move all |x| terms to the RHS, and all integer terms to the LHS.

2 + 12 < 4|x| - |x|

14 < 3|x|

We now divide both sides by 3 to eliminate the coefficient of |x|

14/3 < |x|

or

|x| > 14/3

Now, imagine a number line...

<---- -x ------- 0 ----- x --------->

I would like you to remember the definition of |x| (absolute value) which indicates the distance of a value x from 0. (In other words, it makes any negative number positive and leaves any positive number positive)

This suggests that any x > 14/3 will satisfy |x| > 14/3.

However, we also have to remember that any value inferior to -14/3 will also satisfy the inequality.

As such, we solve the solution as

x > 14/3 OR x < -14/3

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