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Official Solution:


Is \(|x - y| = ||x| - |y||\)?

(1) \(xy > 0\)

This statement implies that \(x\) and \(y\) have the same sign.

If both are positive, \(|x| = x\) and \(|y| = y\), and the question becomes: is \(|x - y| = |x - y|\)? The answer to this question is YES.

If both are negative, \(|x| = -x\) and \(|y| = -y\), and the question becomes: is \(|x - y| = |-x + y|\)? This can be rewritten as: is \(|x - y| = |y - x|\)? Since both \(|x - y|\) and \(|y - x|\) represent the distance between points \(x\) and \(y\) on the number line, they are equal and the answer to this question is once again YES.

Sufficient.

(2) \(x < y < 0\)

This statement implies that both \(x\) and \(y\) are negative. Similarly, as above, the question becomes: is \(|x - y| = |y - x|\)? The answer to this question is YES. Sufficient.


Answer: D
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