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

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==> In the original condition, there are 2 variables (x, y) and in order to match the number of variables to the number of equations, there must be 2 equations. Since there is 1 for con 1) and 1 for con 2), C is most likely to be the answer. By solving con 1) and con 2), from con 1), you get a>-a, 2a>0, or a>0, and from con 2), you get ax>ay, and the inequality sign doesn’t change even if you divide both sides by a because since a>9, you get x>y, hence yes, it is always sufficient.

Therefore, the answer is C.
Answer: C
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Target question: Is x > y?

Statement 1: x + a > x - a
a + a > x - x
2a > 0
a > 0
a is POSITIVE
That's all this statement tells us.
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT.

Statement 2: ax > ay
ax - ay > 0
a(x - y) > 0
Case a: a > 0 and (x - y) > 0, in which case x > y
Case b: a < 0 and (x - y) < 0, 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 a is POSITIVE

Statement 2 tells us that, if a > 0, x > y

Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

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