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Is xy > 0?

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Is xy > 0?

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Is xy > 0?

(1) x - y > -2
(2) x - 2y < -6

The OA is the option C.

How can I get an answer here? I should prove that x and y have the same sign. How can I do that? Experts, may you help me? Thanks in advanced.

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Vincen wrote:
Is xy > 0?

(1) x - y > -2
(2) x - 2y < -6
Statement 1: x > y-2
If y=2 and x= 1, is 1*2 > 0? Yes.
If y= -1 and x = 1, is 1*(-1) > 0? No.
Since the answer can both Yes and No, insufficient.

Statement 2: x < 2y-6
If y=1 and x = -10, is 1*(-10) > 0? No.
If y = 10 and x = 1, is 1*10 > 0? Yes.
Since the answer can be both No and Yes, insufficient.

Statements 1 and 2 combined:
Linking together the two statements, we get:
y-2 < x < 2y-6
y-2 < 2y-6
y > 4.
Since y > 4 and x > y-2, we know that x > 2.
Thus, x and y are both positive.
Sufficient.

The correct answer is C.

Another way to combine the two statements:

Inequalities can be ADDED TOGETHER.
One constraint:
When we add the inequalities together, the <> must face the SAME DIRECTION in each inequality.
Adding together x - y > -2 and -6 > x - 2y, we get:
x - y + (-6) > -2 + x - 2y
- y - 6 > -2 - 2y
y > 4.
Same result as in the solution above.

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Vincen wrote:
Is xy > 0?

(1) x - y > -2
(2) x - 2y < -6

The OA is the option C.

How can I get an answer here? I should prove that x and y have the same sign. How can I do that? Experts, may you help me? Thanks in advanced.
You pick numbers to quickly prove that each statement alone is not sufficient. When testing together, we can multiply the second inequality by -1 to make it -x + 2y > 6. Now we can add them

x - y > -2
-x + 2y > 6
y > 4

So we know y is greater than 4. Now we can add this new inequality to the first one
x - y > -2
y > 4

x > 2

If we know that x and y are both positive, then we know definitively that xy > 0. Together, the statements are sufficient. The answer is C

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