If x and y are integers, what is the value of |x| + |y|?
(1) x + y = 4
(2) |x − y| = 6
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Data InsightsData Sufficiency
Free GMAT Data Sufficiency Practice Question
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Answer & Explanation
Correct answer
C
When x and y have opposite signs, |x| + |y| = |x − y|; when they have the same sign, |x| + |y| = |x + y|.
Statement (1): x + y = 4 leaves |x| + |y| free. (x, y) = (2, 2): |x| + |y| = 4. (x, y) = (6, −2): |x| + |y| = 8. Not sufficient.
Statement (2): |x − y| = 6. (5, −1): opposite signs, |x| + |y| = 6. (7, 1): same signs, |x| + |y| = 8. Not sufficient.
Together: x + y = 4 and |x − y| = 6, so x − y = ±6. Case x − y = 6: x = 5, y = −1. Case x − y = −6: x = −1, y = 5. In both cases x and y have opposite signs (because |x − y| = 6 exceeds |x + y| = 4, which forces one positive and one negative), so |x| + |y| = |x − y| = 6 in every case. Unique value 6. Sufficient together; neither alone.
The same-sign / opposite-sign branch trap. Each statement alone admits both a same-sign and an opposite-sign solution, giving 8 or 6. The combined fact |x − y| = 6 > |x + y| = 4 forces opposite signs (the gap exceeds the sum), collapsing both branches to 6. A solver who picks D or B has not run the branch check that the combination is what eliminates the same-sign case.