Intuitively, when finding the PDF of a RV that is a function of another RV(s), how do I know when to consider which one of the random variables is $>0$ and $<0$.
For example, if $Y = aX + b$, to find the PDF of Y, I need to consider when $a > 0$ and $a < 0$, and find the CDF (and subsequently the PDF) accordingly.
Another example, if $Z = X/Y$, then to find the PDF of Z, I need to consider when $z < 0$ and when $z > 0$, which results in 2 regions for $y > 0$ and $y < 0$.
I would typically just find $P(Y \le y)$ and $P(Z \le z)$ directly without knowing to consider the signs of $a$ and $z, y$ in the 2 examples above, because it's not obvious to me that I need to do that. I wouldn't think to consider the positive/negative regions, let alone know to look at the signs of $a$ or $z, y$.
Is there any way for me to gain this kind of intuition, or if there's some systematic way that I can check?