How do you solve #(x+5)/(x+2)<0#?
A very stodgy mechanical way of looking at this is to say that the expression will be negative if either (x+5) < 0 or (x+2) < 0, but not both
so we look at these pairs
(x+5) < 0 and (x+2) > 0
this requires x < -5 and x > -2, so no solution
(x+5) > 0 and (x+2) < 0
this requires x > -5 and x < -2, so this solution works
further refining this approach, if x = 5, then the numerator is zero, not <0. so we must exclude x = 5
if x = -2, we have a singularity
so the complete answer appears to be
the obvious temptation here must to be to cross multiply ie to say that
but that doesn't work with inequalities. worth thinking about.