The value of x must be greater than *would appreciate thought process as well*

Check the picture below.

notice the first case, the red triangle, if we lay the side 12 flat "almost" on top of the side 15, that will simply make x = 15 - 12, or just 3.  However, we can't lay the side like so, because if we do, what we have is just three pancaked lines making a thick flat-line, and the triangle goes poof.  So whatever "x" is, it must be greater than 3, cannot be 3 though.

and if you look at the green triangle, if we instead lay the side 12 the other way, "almost flat on the ground, then "x" becomes 12 + 15, or 27, but again, we can't do that because we again end up with a flat-line and lose the triangle.

so, whatever "x" is, it has to be 3 < x < 27.

Wow. that is a really cool question.