Strategy: Try to find exceptions for each option!!

A. \(xy>-2\)
--> x = 10, y = -0.5 xy = -5 (<-2)
-- NO

B. \(-x<2y\)
--> Possible values of 2y are -1.9, -1, 0, 2, 5, 100;
Possible values of -x are -2, -2.5, -3, -4 etc
-- YES

C. \(xy<-2\)
--> x = 3, y = -0.5 xy = -1.5 (>-2)
-- NO

D. \(-x>2y\)
--> Possible values of 2y are -1.9, -1, 0, 2, 5, 100;
Possible values of -x are -2, -2.5, -3, -4 etc
-- NO

E. \(x<-2y\)
--> x = 100, y = 0
--> (x, -2y) = (100, 0)
-- NO

IMO Option B

Pls Hit Kudos if you like the solution

The product xy can be anything, because x can be 1 trillion, say, and y can be negative (so xy can be very very negative) or positive (so xy can be very very positive). So A and C do not need to be true. If you then notice that D and E say the same thing (multiply D by -1 on both sides, reversing the inequality when you do, and you get E) then you can be sure neither of them can be the right answer, because if one were always true, the other would be too, and then the question would have two different right answers, which is impossible. So the answer has to be B.

We can prove that though: if x > 2, then -x < -2. If y > -1, then 2y > -2. So putting these two inequalities together, -x < -2 < 2y, and -x < 2y.
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