Why would we need to go through the longer discussion if this is the case?
It's a matter of style, some people like cases, some people like doing them all at once, it's a matter of preference. Think of it as texting someone versus calling them. Both can convey information, but you make a choice as to which you want to use at the time, as doing both would certainly be wasteful.
Another reason is that splitting into cases works for many other inequalities, such as
) for some functions p,q,r, where there isn't a way to do them all at once without introducing erroneous solutions that you will need to investigate later.