Update: The suggested Answer didn't solve the problem
Note: This question can be solved without knowing probability at all. (all you need is the bold text)
Today my lecturer wrote the following on board:
where the text in red specefies for which values the multiplication under integral sign isn't zero.
I understand that we need to split into conditions:
where $\max\{0,z+a\}=0$ which means: $z<-a$
where $\max\{0,z+a\}=z+a$ which means: $z>-a$
but in the first condition where did we get $-b<=z$ from? I understand that without it something will be wrong since for all values the integral will not be 0 (while we know from the text in red that for some it's zero for sure) but I don't understand where it specifically came from...
I have been thinking on this for hours.
It looks like $Y \sim \text{Exp}(\lambda)$ and $X \sim \text{Unif}(a,b)$. In that case $Y \geq 0$ and $a \leq X \leq b$. So $Y-X \geq -b$. Thus $f_{Y-X}(z)=0$ when $z <-b$.
Another way to think about it is combining the inequality $y \geq 0$ with the inequality $z+b \geq y$ gives $z+b \geq 0$, so $z \geq -b$.