Data Sufficiency

[GMAT math practice question]

Is x > 0?

1) |x| + |y| > |x + y|
2) |y| > y

Max@Math Revolution wrote:[GMAT math practice question]

Is x > 0?

1) |x| + |y| > |x + y|
2) |y| > y

Note:
|a|² = a²
|a+b|² = (a+b)²

Statement 1:
Since an absolute value cannot be negative, both sides of the inequality must be nonnegative, allowing us to safely square it:
(|x| + |y|)² > |x + y|²
|x|² + |y|² + 2|x||y| > (x+y)²
x² + y² + 2|x||y| > x² + y² + 2xy
|x||y| > xy
The resulting inequality requires that x and y have DIFFERENT SIGNS.
Thus:
If y<0, then x>0, with the result that the answer to the question stem is YES.
If y>0, then x<0, with the result that the answer to the question stem is NO.
INSUFFICIENT.

Statement 2:
The inequality requires that y<0.
No information about x.
INSUFFICIENT.

Statements combined:
Since y<0 and x and y must have different signs, x>0.
Thus, the answer to the question stem is YES.
SUFFICIENT.

The correct answer is C.

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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

Since we have 2 variables (x and y) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2):

Condition 1) is equivalent to the requirement that xy < 0 and condition 2) is equivalent to the requirement that y < 0. Thus, both conditions together are sufficient.

Since this question is an inequality question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B.

Condition 1)
If x = 1, y = -1, then x > 0 and the answer is 'yes'.
If x = -1, y = 1, then x < 0 and the answer is 'no'.
Condition 1) is not sufficient, since it doesn't yield a unique solution.

Condition 2)
Since condition 2) doesn't provide any information about x, it is not sufficient.

Therefore, C is the answer.
Answer: C

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.