Where is the error? Determining why a proof is incorrect. Why is the following proof incorrect?
I have tried to find out why for a few  hours...
aproof http://forosdelecuador.com/proof.png
The answer is:
We aren't really proving the conclusion of each case is valid for the other case.
Consider the case assumptions for each case to define the real boundaries for x.

From case 1 and case 2, you are using the following logic:
If $x<6$ or $0 < x$, then $0<x<6$.
What you need to do is to deduce from case 1:
If $x-3 \ge 0$ then $3 \le x < 6$.
Similarly for case 2.  Then you should use the logic:
If $3 \le x<6$ or $0< x < 3$, then $0<x<6$.
