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Official Solution:
Is \(x > y\)?
(1) \(x^2 < y^2\)
Taking the square root of the inequality gives \(|x| < |y|\), which implies that \(y\) is further from 0 than \(x\). This is not sufficient to determine whether \(x > y\). For instance, consider \(x = 1\) and \(y = 2\), and \(x = 1\) and \(y = -2\).
(2) \(y < 0\)
This statement is clearly insufficient, since no information about \(x\) is given.
(1) + (2) From (2), we know that \(y\) is negative. From (1), we know that \(y\) is further from 0 than \(x\). If \(x < y\) were true, it would imply that \(x\) is to the left of \(y\) on the number line, making \(x\) further from 0 than \(y\). However, this contradicts the information from (1). Therefore, the only possible scenario is that \(x > y\). Sufficient.
Answer: C
Is \(x > y\)?
(1) \(x^2 < y^2\)
Taking the square root of the inequality gives \(|x| < |y|\), which implies that \(y\) is further from 0 than \(x\). This is not sufficient to determine whether \(x > y\). For instance, consider \(x = 1\) and \(y = 2\), and \(x = 1\) and \(y = -2\).
(2) \(y < 0\)
This statement is clearly insufficient, since no information about \(x\) is given.
(1) + (2) From (2), we know that \(y\) is negative. From (1), we know that \(y\) is further from 0 than \(x\). If \(x < y\) were true, it would imply that \(x\) is to the left of \(y\) on the number line, making \(x\) further from 0 than \(y\). However, this contradicts the information from (1). Therefore, the only possible scenario is that \(x > y\). Sufficient.
Answer: C
