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Official Solution:
Is |x| + x > |y| + y ?
(1) \(xy > 0\)
This statement implies that \(x\) and \(y\) have the same sign.
If both are negative, \(|x| = -x\) and \(|y| = -y\), and the question becomes: is \(-x + x > -y + y\)? This further translates to: is \(0 > 0\)? The answer to this question is NO.
If both are positive, \(|x| = x\) and \(|y| = y\), and the question becomes: is \(x + x > y + y\)? This further translates to: is \(x > y\)? The answer to this question can be both YES and NO.
Not sufficient.
(2) \(x + y < 0\)
This statement is clearly insufficient. For instance, consider \(x = 0\) and \(y = -1\) for a NO answer, and \(x = 1\) and \(y = -2\) for a YES answer.
(1)+(2) From (1) we have that \(x\) and \(y\) have the same sign, and from (2) we have that the sum of \(x\) and \(y\) is negative. Hence, both \(x\) and \(y\) must be negative. In (1), we deduced that if both are negative, the question becomes whether \(0 > 0\), the answer to which is NO. Sufficient.
Answer: C
Is |x| + x > |y| + y ?
(1) \(xy > 0\)
This statement implies that \(x\) and \(y\) have the same sign.
If both are negative, \(|x| = -x\) and \(|y| = -y\), and the question becomes: is \(-x + x > -y + y\)? This further translates to: is \(0 > 0\)? The answer to this question is NO.
If both are positive, \(|x| = x\) and \(|y| = y\), and the question becomes: is \(x + x > y + y\)? This further translates to: is \(x > y\)? The answer to this question can be both YES and NO.
Not sufficient.
(2) \(x + y < 0\)
This statement is clearly insufficient. For instance, consider \(x = 0\) and \(y = -1\) for a NO answer, and \(x = 1\) and \(y = -2\) for a YES answer.
(1)+(2) From (1) we have that \(x\) and \(y\) have the same sign, and from (2) we have that the sum of \(x\) and \(y\) is negative. Hence, both \(x\) and \(y\) must be negative. In (1), we deduced that if both are negative, the question becomes whether \(0 > 0\), the answer to which is NO. Sufficient.
Answer: C
