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Is \(|x - y| \gt |x + y|\)?
(1) \(x^2 - y^2 = 9\)
(2) \(x - y = 2\)
(1) \(x^2 - y^2 = 9\)
(2) \(x - y = 2\)
07 Dec 2014, 15:37
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Official Solution:
Is \(|x - y| \gt |x + y|\)?
Square both sides:
Is \((|x - y|)^2 \gt (|x + y|)^2\)?
Is \(x^2 - 2xy +y^2 > x^2+2xy+y^2\)?
Is \(0>4xy\)?
Is \(0>xy\)?
Therefore, the question essentially asks whether \(x\) and \(y\) have different signs.
(1) \(x^2 - y^2 = 9\).
Can we conclude from this whether \(x\) and \(y\) have different signs? No. Among infinitely many other solutions, we could have that \(x\) is 5 and \(y\) is 4 OR \(x\) is -5 and \(y\) is 4. Not sufficient.
(2) \(x - y = 2\).
This statement also does not help. Consider \(x=4\) and \(y = 2\) OR \(x = 1\) and \(y = -1\). Not sufficient.
(1)+(2) From (1), we have \((x-y)(x + y) = 9\), and from (2), we have \(x - y = 2\). Substituting the value of \(x - y\) into \((x-y)(x + y) = 9\) results in \(2(x + y) = 9\), implying that \(x + y = 4.5\). Thus, \(|x + y| = 4.5\) and \(|x - y| = 2\) (from 2). The question of whether \(|x - y| > |x + y|\) can now be answered: NO, as \(|x - y| = 2\) is NOT greater than \(|x + y| = 4.5\). Sufficient.
Answer: C
Is \(|x - y| \gt |x + y|\)?
Square both sides:
Is \((|x - y|)^2 \gt (|x + y|)^2\)?
Is \(x^2 - 2xy +y^2 > x^2+2xy+y^2\)?
Is \(0>4xy\)?
Is \(0>xy\)?
Therefore, the question essentially asks whether \(x\) and \(y\) have different signs.
(1) \(x^2 - y^2 = 9\).
Can we conclude from this whether \(x\) and \(y\) have different signs? No. Among infinitely many other solutions, we could have that \(x\) is 5 and \(y\) is 4 OR \(x\) is -5 and \(y\) is 4. Not sufficient.
(2) \(x - y = 2\).
This statement also does not help. Consider \(x=4\) and \(y = 2\) OR \(x = 1\) and \(y = -1\). Not sufficient.
(1)+(2) From (1), we have \((x-y)(x + y) = 9\), and from (2), we have \(x - y = 2\). Substituting the value of \(x - y\) into \((x-y)(x + y) = 9\) results in \(2(x + y) = 9\), implying that \(x + y = 4.5\). Thus, \(|x + y| = 4.5\) and \(|x - y| = 2\) (from 2). The question of whether \(|x - y| > |x + y|\) can now be answered: NO, as \(|x - y| = 2\) is NOT greater than \(|x + y| = 4.5\). Sufficient.
Answer: C
06 Jul 2015, 00:29
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Bunuel
Hi bunuel,
Why can't (1) be sufficient. Here is my reasoning:
(x-y)(x+y)=9 can be rewritten as (5-4)(5+4) or even (4-5)(4+5).
Either ways, the answer to the main question will always yield a 'no'. Doesn't this mean statement 1 is sufficient?
