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Tough DS

by edwardyong » Fri Oct 30, 2009 9:52 am
In which quadrant of the coordinate plane does the point (x, y) lie?

(1) |xy| + x|y| + |x|y + xy > 0
(2) -x < -y < |y|

If someone know how to do this question please explain the step you have taken to solve the question. I will provide OA later. Thanks.
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by ayashlaha » Fri Oct 30, 2009 10:45 am
I think the answer is D

(x,y) can be in one of 4 quadrants , i.e. it may be one of the following
(+,+) (+,-) (-,+) or (-,-)


Let's evaluate statement 1
Substituting each of these combinations in the equation, we only get a non zero result for (+,+) ie the 1st quadrant. All others lead to a zero.
Hence 1st Quadrant SUFFICIENT

Let's evaluate statement 2

Lets consider
-y < |y|
If y=2--> -y=-2 and |y|=2 Satisfies inequality
If y=-2 --> -y=2 and |y|=2 Violates inequality

This imples y is a positive number.

Likewise if
-x<-y

Multiply both sides by -1 and change inequality direction
x>y
implies x is (+)ve.

Hence this makes (x,y) lie in the 1st quadrant SUFFICIENT

Thus the answer should be D

Please post official answer.
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by Testluv » Fri Oct 30, 2009 6:53 pm
In which quadrant of the coordinate plane does the point (x, y) lie?

(1) |xy| + x|y| + |x|y + xy > 0
(2) -x < -y < |y|
Hi edwardyong,

ayashlaha's approach was excellent; I would have done the exact same. Let's just elaborate it a bit more.

Let's start by focussing on the question. It is really just asking us whether we can figure out the positive/negative status of both x and y.

Let's take a look at statement 1. Looks intimidating but can be handled quickly by picking some numbers and being organized.

case 1: both x and y are positive. Let x be +2 and y be +3. But the moment you start plugging in (or before you start plugging in) you will realize that, if they are both positive, the left hand side is clearly positive, and the inequality is satisfied. So they can both be positive.

case 2: both x and y are negative. Now let x be -2 and y be -3. There are four terms on the left hand side in the inequality of statement one. If both x and y are negative, you will quickly see that the "outer" terms (the first and the fourth) are positive. This is because the first term is absolute value and the fourth term is (neg)*(neg).
But you will see that the two "inner" terms are negative. So everything will cancel out and the left hand side will equal zero...but this fails to satisfy the statement. So they can't both be negative.

case 3: one is positive and the other is negative: by plugging in numbers you will again see that the left hand side is zero, thereby failing to satisfy the statement: the first terms is positive, the fourth term is negative, one of the "inner" terms is positive and the other "inner" term is negative. So we can't have one being positive and the other being negative. And if either is zero, the statement is clearly not satisfied.

So the only way to satisfy statement one is if both x and y are positive.

ayashlaha's explanation for statement 2 was excellent so I'll leave it.

Both statements independently sufficient. We should choose D.
Last edited by Testluv on Sun Nov 01, 2009 1:36 pm, edited 2 times in total.
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OA and OE

by edwardyong » Fri Oct 30, 2009 8:10 pm
Thanks to both ayashkaha and testluv.

You guys nailed the question......

The OA is D

The OE:

In essence, this question asks whether the signs of both x and y can be determined.

(1) SUFFICIENT: The key to evaluating this statement is to see which values of x and y actually satisfy it ("crack the code"). To do so, consider all possibilities for the signs of x and y.
"¢ x > 0, y > 0: The left side becomes xy + xy + xy + xy = 4xy, which is a positive number; the statement is satisfied.
"¢ x < 0, y > 0: The left side becomes xy - xy + xy - xy = 0, so the statement is not satisfied.
"¢ x > 0, y < 0: The left side becomes xy + xy - xy - xy = 0, so the statement is not satisfied.
"¢ x < 0, y < 0: The left side becomes xy - xy - xy + xy = 0, so the statement is not satisfied.
"¢ Either x or y (or both) is 0: The left side becomes 0 + 0 + 0 + 0 = 0, so the statement is not satisfied.
Therefore, statement (1) can be rephrased simply as "Both x and y are positive." The point (x, y) is thus in the first quadrant.

(2) SUFFICIENT: If -y does not equal |y|, then y must be positive (and -y must be negative). Since -x < -y, we know that -x is also negative, so x is also positive. The point (x, y) is therefore in the first quadrant.

The correct answer is D.
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by Testluv » Fri Oct 30, 2009 9:21 pm
Hi Edward,

actually the question was phrased "In which quadrant of the coordinate plane does the point (x, y) lie?"

As 0,0 does not lie in any quadrant, I interpreted this as meaning that x,y does lie in a quadrant, which is why I did not consider the possibility of both x and y equalling zero. That is, I read the english in the question as presupposing that x,y lay in a quadrant.

GMAT takes a lot of pains to make sure there is no ambiguity in questions, but if you think there are multiple reasonable interpretations of some english, then just adopt the interpretation that makes the most common sense.

That said, the rule for data sufficiency is: no assumptions. Accordingly, on this question, it would have been safer to consider that x,y could be 0,0. And, interestingly enough, if a statement had yielded a solution of 0,0 and only 0,0, then that statement would have been sufficient becuase the answer to the question would be certain ("x,y does not lie in any quadrant".)
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