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Official Solution:
Is \(x < y\) ?
(1) \(x^3 < y^3\).
Note that we can always take an odd root from both sides of an inequality, just as we can raise both parts of an inequality to an odd power. If we take the 3rd root from \(x^3 < y^3\), we obtain \(x < y\). Sufficient.
(2) \((x+y)(x-y) < 0\).
This statement implies that \(x^2-y^2 < 0\), which means \(x^2 < y^2\). Consequently, \(|x| < |y|\). This indicates that \(y\) is further from 0 than \(x\) is. However, this is not sufficient to determine if \(x < y\). For instance, consider \(x=1\) and \(y=2\) for a YES answer, and \(x=1\) and \(y=-2\) for a NO answer.
Answer: A
Is \(x < y\) ?
(1) \(x^3 < y^3\).
Note that we can always take an odd root from both sides of an inequality, just as we can raise both parts of an inequality to an odd power. If we take the 3rd root from \(x^3 < y^3\), we obtain \(x < y\). Sufficient.
(2) \((x+y)(x-y) < 0\).
This statement implies that \(x^2-y^2 < 0\), which means \(x^2 < y^2\). Consequently, \(|x| < |y|\). This indicates that \(y\) is further from 0 than \(x\) is. However, this is not sufficient to determine if \(x < y\). For instance, consider \(x=1\) and \(y=2\) for a YES answer, and \(x=1\) and \(y=-2\) for a NO answer.
Answer: A
