The easiest way to deal with this task is to selectively look through some numbers for A and B, depending on the conditions.
Quote:
(1) \(a<0\)
This gives us four potential scenarios, based on whether \(a+5\) is still below zero or becomes positive:
- \(a<-5\) and \(b<0\), for instance \(a=-10\) and \(b=-1\)
\(\frac{-5}{-1}\)<\(\frac{-10}{-1}\) - \(-5<a<0\) and \(b<0\), for instance \(a=-3\) and \(b=-1\)
\(\frac{2}{-1}\)<\(\frac{-3}{-1}\) - \(a<-5\) and \(b>0\), for instance \(a=-10\) and \( b=1\)
\(\frac{-5}{1}\)>\(\frac{-10}{1}\) - \(-5<a<0\) and \(b>0\), for instance \(a=-3\) and \(b=1\)
\(\frac{2}{1}\)>\(\frac{-3}{1}\)
So, as we can see, we have both options here - both 'more' and 'less', as highlighted in yellow. Therefore,
Condition 1 by itself is not sufficient.Quote:
(2) \(b<0\)
In this case, because we have sort of 'fixed' B, we need to look into three options for A - below -5, above -5 and above 0:
- \(a<-5\) and \(b<0\), for instance \(a=-10\) and \(b=-1\)
\(\frac{-5}{-1}\)<\(\frac{-10}{-1}\) - \(-5<a<0\) and \(b<0\), for instance \(a=-3\) and \(b=-1\)
\(\frac{2}{-1}\)<\(\frac{-3}{-1}\) - \(0<a\) and \(b<0\), for instance \(a=3\) and \(b=-1\)
\(\frac{8}{-1}\)<\(\frac{3}{-1}\)
So, as can bee seen, ALL the signs here are 'less' which basically means that
Condition 2 by itself is sufficient.Therefore,
the correct answer is B.