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Is \(x < 10 < y\)?
(1) \(x < y\) and \(xy = 100\)
(2) \(x^2 < 100 < y^2\)
(1) \(x < y\) and \(xy = 100\)
(2) \(x^2 < 100 < y^2\)
10 Nov 2021, 03:24
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Bookmarks
Official Solution:
Is \(x < 10 < y\)?
(1) \(x < y\) and \(xy = 100\)
If \(x=-20\) and \(y=-5\), then \(x < y\) but \(10\) is not between them, so the answer is NO. However, if \(x=5\) and \(y=20\), then the answer is YES.
Not sufficient.
(2) \(x^2 < 100 < y^2\)
The first inequality, \(x^2 < 100\), implies \(-10 < x < 10\). So, we are left to establish whether \(y > 10\).
The second inequality, \(100 < y^2\), implies \(y > 10\) or \(y < -10\).
So, we can have the following two cases:
a. If \(y > 10\), then \(x < 10 < y\) is true.
b. If \(y < -10\), then \(x > y\), and therefore \(x < 10 < y\) is false.
Not sufficient.
(1)+(2) Since from (1) we have that \(x < y\), then case b from (2), saying that \(x > y\), cannot be true, hence we are left with case a, and therefore, \(x < 10 < y\). Sufficient.
Answer: C
Is \(x < 10 < y\)?
(1) \(x < y\) and \(xy = 100\)
If \(x=-20\) and \(y=-5\), then \(x < y\) but \(10\) is not between them, so the answer is NO. However, if \(x=5\) and \(y=20\), then the answer is YES.
Not sufficient.
(2) \(x^2 < 100 < y^2\)
The first inequality, \(x^2 < 100\), implies \(-10 < x < 10\). So, we are left to establish whether \(y > 10\).
The second inequality, \(100 < y^2\), implies \(y > 10\) or \(y < -10\).
So, we can have the following two cases:
a. If \(y > 10\), then \(x < 10 < y\) is true.
b. If \(y < -10\), then \(x > y\), and therefore \(x < 10 < y\) is false.
Not sufficient.
(1)+(2) Since from (1) we have that \(x < y\), then case b from (2), saying that \(x > y\), cannot be true, hence we are left with case a, and therefore, \(x < 10 < y\). Sufficient.
Answer: C
