Is xy > 0? (1) x – y > –5 (2) x – 2y < –7

Great question!

Target question: Is xy > 0

Statement 1: x – y > –5
This statement doesn't FEEL sufficient, so I'll TEST some values.
There are several values of x and y that satisfy statement 1. Here are two:
Case a: x = 1 and y = 1, in which case xy = (1)(1) = 1. So, xy > 0
Case b: x = 1 and y = -1, in which case xy = (1)(-1) = -1. So, xy < 0
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Aside: For more on this idea of plugging in values when a statement doesn't feel sufficient, you can read my article: https://www.gmatprepnow.com/articles/dat ... lug-values

Statement 2: x – 2y < –7
There are several values of x and y that satisfy statement 1. Here are two:
Case a: x = -10 and y = -1, in which case xy = (-10)(-1) = 10. So, xy > 0
Case b: x = -10 and y = 1, in which case xy = (-10)(1) = -10. So, xy < 0
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that x – y > –5, which we can rewrite as -5 < x - y
Statement 2 tells us that x – 2y < –7
Since -7 < -5, we can COMBINE the inequalities to get: x – 2y < –7 < -5 < x - y
Focus on x – 2y < x - y
Subtract x from both sides: -2y < -y
Add 2y to both sides: 0 < y
So, y is POSITIVE, but there's no info about x.
Consider these two conflicting scenarios:
Case a: x = 2 and y = 5, in which case xy = (2)(5) = 10. So, xy > 0
Case b: x = -1 and y = 3.5, in which case xy = (-1)(3.5) = -3.5. So, xy < 0
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

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