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16 Sep 2014, 00:15
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If the product of two integers \(x\) and \(y\) is negative, what is the value of \(x - y\)?
(1) \(x + y = 2\).
(2) \(-3 < x < y\).
(1) \(x + y = 2\).
(2) \(-3 < x < y\).
16 Sep 2014, 00:15
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Official Solution:
If the product of two integers \(x\) and \(y\) is negative, what is the value of \(x - y\)?
Note that \(xy < 0\) implies that \(x\) and \(y\) have different signs.
(1) \(x + y = 2\).
There are infinitely many sets of (x, y) that can satisfy this condition, leading to different values for \(x - y\). For instance, if \(x = 3\) and \(y = -1\), then \(x - y = 4\). However, if \(x = 4\) and \(y = -2\), then \(x - y = 6\). Not sufficient.
(2) \(-3 < x < y\).
Given that \(x\) and \(y\) must have different signs, it follows that \(-3 < x < 0 < y\), which means that \(x\) must be negative and \(y\) must be positive. Therefore, \(x\), being an integer, can only be -2 or -1. However, \(y\) can be any positive integer. Not sufficient.
(1)+(2) If from (2) \(x = -2\), then from \(x + y = 2\), we deduce that \(y\) equals 4, resulting in the value of \(x - y\) being -6. Conversely, if from (2) \(x = -1\), then from \(x + y = 2\), we deduce that \(y\) equals 3, resulting in the value of \(x - y\) being -4. Not sufficient.
Answer: E
If the product of two integers \(x\) and \(y\) is negative, what is the value of \(x - y\)?
Note that \(xy < 0\) implies that \(x\) and \(y\) have different signs.
(1) \(x + y = 2\).
There are infinitely many sets of (x, y) that can satisfy this condition, leading to different values for \(x - y\). For instance, if \(x = 3\) and \(y = -1\), then \(x - y = 4\). However, if \(x = 4\) and \(y = -2\), then \(x - y = 6\). Not sufficient.
(2) \(-3 < x < y\).
Given that \(x\) and \(y\) must have different signs, it follows that \(-3 < x < 0 < y\), which means that \(x\) must be negative and \(y\) must be positive. Therefore, \(x\), being an integer, can only be -2 or -1. However, \(y\) can be any positive integer. Not sufficient.
(1)+(2) If from (2) \(x = -2\), then from \(x + y = 2\), we deduce that \(y\) equals 4, resulting in the value of \(x - y\) being -6. Conversely, if from (2) \(x = -1\), then from \(x + y = 2\), we deduce that \(y\) equals 3, resulting in the value of \(x - y\) being -4. Not sufficient.
Answer: E
General Discussion
03 Jun 2023, 13:53
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
