Baten80 wrote:
Is xy > 0?
(1) x - y > -2
(2) x - 2y < -6
We need to determine whether the product of x and y is positive. We should recall that the product of two numbers is positive only if both the numbers are positive or if both are negative.
Statement One Alone:
x - y > -2
Statement one tells us that the difference between x and y is -2; it does not tell us anything about the signs of x and y. For instance, if x = 2 and y = 1, we have x - y = 1 > -2, and xy is positive. However, if x = 3 and y = -2, 3 - (-2) = 5 > -2, but xy is negative. Statement one alone is not sufficient. We can eliminate answer choices A and D.
Statement Two Alone:
x - 2y < -6
Again, we have a statement that tells us nothing about the signs of x and y. For instance, if x = 3 and y = 5, then x - 2y = 3 - 2(5) = 3 - 10 = -7 < -6, and xy is positive. However, if x = -1 and y = 5, then x - 2y = -1 - 2(5) = -11 < -6, and xy is negative. Statement two alone is not sufficient. We can eliminate answer choice B.
Statements One and Two Together:
Let’s manipulate the first inequality to read: y < x + 2. Similarly, we can manipulate the second inequality to read: y > (1/2)x + 3.
Thus, we can say the following:
(1/2)x + 3 < y < x + 2
(1/2)x + 3 < x + 2
x + 6 < 2x + 4
2 < x
Thus, x is positive.
We also know the following:
y > (1/2)x + 3
Since x is greater than two, let’s see what we can determine about y, if we substitute 2 for x.
y > (1/2)(2) + 3
y > 4
So y is positive as well. Both statements together are sufficient.
Answer: C
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