Double sided modulus equation: $|2x-1| = |3x+2|$ Considering this equation $$|2x-1| = |3x+2|$$
My question is what is the reasoning behind taking only the positive and negative values of one side of the equation to solve the problem. I have seen many people do that and it seems to work. 
For e.g., $$2x-1 = 3x+2$$ 
          $$2x-1 = -(3x+2),$$ 
then solve for $x$.On the other hand, I have been taught that I should be testing both positive and negative values on both sides which will eventually give 4 cases to solve instead of 2. This makes more sense since disregarding the absolute value sign give rise to the situation, a negative and positive value.

In this example you actually can reduce this to two cases.  Either the expressions inside the absolute values share the same sign, or they have opposite signs.
If you assume the left is negative and the right is positive.
$-(2x-1) = 3x + 2$
And the case the the right is positive and the left is negative.
$2x -1 = -(3x+2)$
You will get the same solutions.  However, for one case, the results are in contradiction to the assumption.
If you have a more if you added in something not inside the absolute value brakets... e.g. $|x+1| - |2x-1| = 1$
You should be testing all 4 cases.  Or  $x<-1, -1<x<\frac 12, \frac 12<x$ will suffice.
