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Official Solution:
Is \(x > y\)?
(1) \(x + y > 0\)
This statement tells us that the sum of two numbers is greater than 0. This is clearly insufficient to determine which one of them is greater. Not sufficient.
(2) \(y^2 > x^2\)
Taking the square root of the inequality gives \(|y| > |x|\), 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 \(y = -2\).
(1) + (2) The second statement implies that \((y - x)(y + x) > 0\). Since from the first statement we know that \(x + y\) is positive, then \(y - x\) must also be positive, which implies that \(y > x\), giving a NO answer to the question. Sufficient.
Answer: C
Is \(x > y\)?
(1) \(x + y > 0\)
This statement tells us that the sum of two numbers is greater than 0. This is clearly insufficient to determine which one of them is greater. Not sufficient.
(2) \(y^2 > x^2\)
Taking the square root of the inequality gives \(|y| > |x|\), 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 \(y = -2\).
(1) + (2) The second statement implies that \((y - x)(y + x) > 0\). Since from the first statement we know that \(x + y\) is positive, then \(y - x\) must also be positive, which implies that \(y > x\), giving a NO answer to the question. Sufficient.
Answer: C
