Is ab > 0?

Note that:
You can only add inequalities when their signs are in the same direction:

If \(a>b\) and \(c>d\) (signs in same direction: \(>\) and \(>\)) --> \(a+c>b+d\).
Example: \(3<4\) and \(2<5\) --> \(3+2<4+5\).

You can only apply subtraction when their signs are in the opposite directions:

If \(a>b\) and \(c<d\) (signs in opposite direction: \(>\) and \(<\)) --> \(a-c>b-d\) (take the sign of the inequality you subtract from).
Example: \(3<4\) and \(5>1\) --> \(3-5<4-1\).

Back to the original question:

Is ab > 0?

Question basically asks whether \(a\) and \(b\) have the same sign.

(1) a – b > 0 --> \(a>b\). Not sufficient to say whether \(a\) and \(b\) have the same sign.

(2) a + b <0 --> \(a<-b\). Again not sufficient to say whether \(a\) and \(b\) have the same sign.

(1)+(2) subtract (2) from (1): \((a-b)-(a+b)>0\) --> \(b<0\) --> but \(a\) could still be positive or negative (or even zero), for example: \(a=-1\) and \(b=-2\) or \(a=1\) and \(b=-2\). Not sufficient.

Or: as from (1) \(b<a\) and from (2) \(a<-b\) then \(b<a<-b\) --> \(a<|b|\) --(b)-----0-----(-b)-- --> \(a\) is somewhere between \(b\) and \({-b}\) so it can be positive, negative or zero). Not sufficient.

Answer: E.