Is xy > 0?

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Is xy > 0?

by pims » Wed Jul 09, 2014 8:20 am
Is xy > 0?

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

OA: C

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by [email protected] » Wed Jul 09, 2014 8:43 am
Hi pims,

This DS question is perfect for TESTing VALUES.

We're asked if XY > 0? This is a YES/NO question.

Fact 1: X - Y > - 2

This can rewritten as:

X + 2 > Y

If X = 1, Y = 1, XY = 1 and the answer to the question is YES
If X = 1, Y = 0, XY = 0 and the answer to the question is NO
Fact 1 is INSUFFICIENT

Fact 2: X - 2Y < -6

This can be rewritten as:

X + 6 < 2Y

If X = 0, Y = 4, then XY = 0 and the answer to the question is NO
If X = 1, Y = 4, then XY = 4 and the answer to the question is YES
Fact 2 is INSUFFICIENT

Combined, we know:
X + 2 > Y
X + 6 < 2Y

2Y - 6 > X > Y - 2

From this, with a bit of "tinkering", we can deduce that...

Y CANNOT be 0, since -6 > X > -2 is impossible
Y CANNOT be negative (it would create the same "impossible" situation)
Y MUST be > 4

Since Y > 4, then X > 2

This means that X and Y are BOTH POSITIVE, so the answer to the question is ALWAYS YES.
Combined, SUFFICIENT

Final Answer: C

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by pims » Wed Jul 09, 2014 8:53 am
Thanks Rich,

I had the same solution as you for A and B separately. I just don't know how to approach the combined statements.

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by GMATGuruNY » Wed Jul 09, 2014 9:23 am
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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by Brent@GMATPrepNow » Wed Jul 09, 2014 1:51 pm
pims wrote:Is xy > 0?

(1) x - y > -2
(2) x - 2y < -6
Target question: Is xy > 0?

Statement 1: x - y > -2
There are several values of x and y that satisfy this condition. Here are two:
Case a: x = 1 and y = 1, in which case xy = (1)(1) = 1. Here, xy > 0
Case b: x = 1 and y = -1, in which case xy = (1)(-1) = -1. Here, xy < 0
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x - 2y < -6
There are several values of x and y that satisfy this condition. Here are two:
Case a: x = 1 and y = 10, in which case xy = (1)(10) = 10. Here, xy > 0
Case b: x = -10 and y = 1, in which case xy = (-10)(1) = -10. Here, xy < 0
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1: x - y > -2
Statement 2: x - 2y < -6

IMPORTANT: We can solve this system of inequalities (through addition) as long as we get the INEQUALITIES FACING THE SAME DIRECTION first.

Keep statement 1 as is: x - y > -2
Multiply statement 2 by -1 to get: -x + 2y > 6
ADD the inequalities to get: y > 4
In other words, y IS POSITIVE

-------

Let's do this one more time.
Statement 1: x - y > -2
Statement 2: x - 2y < -6

Multiply statement 1 by 2 to get: 2x - 2y > -4
Multiply statement 2 by -1 to get: -x + 2y > 6
ADD the inequalities to get: x > 2
In other words, x IS POSITIVE

If x and y are BOTH POSITIVE, we can be certain that xy > 0
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C

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Brent
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