sum of random variables - help to evaluate support of function

Question.

otherwise

Suppose Z=X+Y. Find the density function of Z.

Answer.

I am having difficulty understanding how to work out the support for Z. I guess I would try to make u and (z-u) non-negative, right?

For example, I know that

Then if X=u then Y=z-u and in this case

Where to from here?... split the z interval into ? (just looking at similar questions, they do split z into several intervals, but I haven't figured out how they decide which ones).