Is x > y?

[BTGModeratorVI]

Is x > y?

1) x + a > x - a
2) ax > ay

Answer: C
Source: Math Revolution

[Brent@GMATPrepNow]

Is x > y?

1) x + a > x - a
2) ax > ay

Answer: C
Source: Math Revolution

Target question: Is x > y?

Statement 1: x + a > x - a
This statement doesn't include any information about y, so there's no way to answer the target question.
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: ax > ay
Some students will divide both sides by a and incorrectly conclude that x > y.
However, before we divide by a variable, we must ensure that the variable is EITHER positive OR negative, because if we divide by a negative value, we must reverse the direction of the inequality, and if we divide by a positive value, the direction of the inequality stays the same. As it stands, we don't know whether a is positive or negative.

To see what I mean, consider these values of a, x and y that satisfy the given condition:
Case a: a = 1, x = 3 and y = 2, in which case x > y
Case b: a = -1, x = 2 and y = 3, in which case x < y
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 + a > x - a
Add a to both sides to get: x + 2a > x
Subtract x from both sides to get: 2a > 0
Divide both sides by 2 to get: a > 0. In other words, a is POSITIVE

Statement 2 tells us that ax > ay
Now that we know that a is POSITIVE, we can take ax > ay and safely divide both sides by a to get: x > y
PERFECT!
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C