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Source: — Data Sufficiency |
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by GMATGuruNY » Fri Jan 02, 2015 1:02 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 is YES in the first case but NO in the second case, 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 is NO in the first case but YES in the second case, INSUFFICIENT.

Statements 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, with the result that xy > 0.
SUFFICIENT.

The correct answer is C.
Last edited by GMATGuruNY on Tue Feb 03, 2015 6:57 am, edited 2 times in total.
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by Brent@GMATPrepNow » Fri Jan 02, 2015 8:22 am
Is xy > 0 ?

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

Statement 1: x-y > -2
There are several pairs of numbers that meet this condition. Here are two:
Case a: x = 5 and y = 1, in which case xy is greater than 0
Case b: x = 5 and y = -1, in which case xy is not greater than 0
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x - 2y < -6
There are several pairs of numbers that meet this condition. Here are two:
Case a: x = 1 and y = 5, in which case xy is greater than 0
Case b: x = -1 and y = 5, in which case xy is not greater than 0
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined:
Here's what we know:
x-y > -2
x-2y < -6

Since both inequalities have an x, let's isolate x in both of them to get:
y-2 < x
x < 2y-6

Aside: Notice that I rewrote them so that the 2 inequality symbols are pointing in the same direction.

Now we can combine these inequalities to get: y-2 < x < 2y-6
Next, remove the x to get: y-2 < 2y-6
Then subtract y from both sides and add 6 to both sides to get: 4 < y
Great, we now know that y is positive.
Also, if y-2 < x (and y>4), then we know that x must also be positive
Since we now know that x and y are positive, we can be certain that xy is greater than 0

So, the answer is C

Cheers,
Brent
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by ceilidh.erickson » Sat Jan 03, 2015 6:42 pm
Mitch, I think you accidentally posted the solution to the wrong question!

I'll add 2 things to Brent's great explanation:

a) As I'm sure you know, any time you see an inequality relative to 0, frame your question in terms of positives and negatives.
Is xy > 0?
Rephrased question: do x and y have the same sign?

Brent gave good examples of testing numbers to prove that x and y could have the same sign or different signs with each statement individually.

b) If you'd prefer not to isolate x first, you can combine the inequalities by simply flipping the second one around:
x - 2y < -6 --> -6 > x - 2y

Now, line up the inequalities and you can add them together:

x - y > -2
-6 > x - 2y
_______________
x - y - 6 > x - 2y - 2

Simplify: -y - 6 > -2y - 2
y - 6 > -2
y > 4

Substitute:
x - (grt 4) > -2
x > 2

x and y are both positive, so together the statements are sufficient.
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by GMATGuruNY » Sat Jan 03, 2015 6:55 pm
ceilidh.erickson wrote:Mitch, I think you accidentally posted the solution to the wrong question!
Thanks for pointing this out, Ceilidh.
My post above now offers a solution to the correct question.
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by J N » Fri Jan 23, 2015 3:50 pm
but if substitute in 2nd equation does not hold?

y>4

-6 > x - 2y

-6> x - 2(grt4)

-6> x -grt8

grt 2 > x

x could be 0 or neg ?
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by Matt@VeritasPrep » Sun Jan 25, 2015 8:26 pm
J N, I'd think of it this way.

x - y > -2, or x > y - 2.

Since y > 4, y - 2 > 2.

We know that x > y - 2, so x > 2.
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