GMAT3

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[Brent@GMATPrepNow]

Is xy > 0?

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

Oops, looks like I transcribed the question incorrectly (as Rich notes below).
I've edited my response accordingly.

Target question: Is xy > 0?

Statement 1: x - y > -2
Does this provide enough information to answer the target question with CERTAINTY? No.
Consider these two possible values of x and y that satisfy the given inequality:
Case a: x = 1, y = 1, in which case xy > 0
Case b: x = -1, y = 0.5, in which case xy < 0
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x - 2y < -6
Consider these two possible values of x and y that satisfy the given inequality:
Case a: x = 1, y = 5, in which case xy > 0
Case b: x = -1, y = 5, in which case xy < 0
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that x - y > -2
Statement 2 tells us that x - 2y < -6

IMPORTANT: If we have two inequalities facing the same direction, we can ADD them together to create a new inequality and, if we're lucky, eliminate a variable.

So, take the inequality in statement 2, x - 2y < -6, and multiply both sides of the inequality by -1.
We get: -x + 2y > 6
Now take x - y > -2 and add the inequalities to get...
y > 4. GREAT, this means that y MUST BE POSITIVE
Can we use the fact that y > 4 to conclude anything about x? Yes.
If we take statement 1, x - y > -2, and add y to both sides, we get: x > y - 2
Now that we know y > 4, we can be conclude that x > 2. This means that x MUST BE POSITIVE as well.
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

Cheers,
Brent

[[email protected]]

Hi mukherjee.tanu3,

**I think that Brent mis-copied Fact 2, so his answer will be different from the Official answer**

This DS question can be solved with Number Properties.

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

Here are the Number Properties you need to know to answer this question:

(Positive)(Positive) = Positive
(Negative)(Negative) = Positive
(Positive)(Negative) = Negative
(0)(Anything) = 0

Fact 1: X - Y > -2

With a bit of algebra, we have...

X + 2 > Y

If X and Y are BOTH POSITIVE, then the answer to the question is YES.
If either X or Y is ZERO, then the answer to the question is NO.
Fact 1 is INSUFFICIENT

Fact 2: X - 2Y < -6

With a bit of algebra, we have...

X + 6 < 2Y

If X and Y are BOTH POSITIVE, then the answer to the question is YES.
If either X or Y is ZERO, then the answer to the question is NO.
Fact 2 is INSUFFICIENT

Combined, we have an interesting limitation...
X + 2 > Y
X + 6 < 2Y

I'm going to combine these two inequalities into one big inequality...
X > Y - 2
X < 2Y - 6

2Y - 6 > X > Y - 2

With this big inequality, the limitations are easy to define...

2Y - 6 > Y - 2
Y > 4

Since Y must be greater than 4, we know that X MUST be POSITIVE.
Both X and Y MUST be POSITIVE, so the answer to the question is ALWAYS YES.
Combined, SUFFICIENT

Final Answer: C

GMAT assassins aren't born, they're made,
Rich

[GMATGuruNY]

Is xy > 0?

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

It can be helpful to put one variable IN TERMS OF THE OTHER.

Statement 1: x - y > -2
Thus, x > y-2.
It's possible that y=1 and x=0, in which case xy=0.
It's possible that y=1 and x=1, in which case xy>0.
INSUFFICIENT.

Statement 2: x-2y < -6
x < 2y-6.
It's possible that y=0 and x=-7, in which case xy=0.
It's possible that y=-1 and x=-10, in which case xy>0.
INSUFFICIENT.

Statements combined:
Inequalities with two variables can be ADDED TOGETHER.
One constraint: the < > must face the SAME DIRECTION in each inequality.
Here, before adding, we can rephrase x < 2y-6 as 2y-6 > x.
Adding together x > y-2 and 2y-6 > x, we get:
x + 2y - 6 > y - 2 + x
2y - 6 > y - 2
y > 4.
Since y > 4 and x > y-2, x > 2.
Thus, xy > 0.

The correct answer is C.