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Is xy > 0 ?

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Is xy > 0 ?

by sachin_yadav » Tue Feb 03, 2015 5:07 am
Is xy > 0 ?
(1) x - y > -2
(2) x - 2y < -6

OA is C

[spoiler]It took me 2 mins 47 secs. My first approach was using numbers of different signs and then solving two simultaneous equations in the end. Is there any other faster approach ?

[/spoiler]

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by MartyMurray » Tue Feb 03, 2015 5:46 am
sachin_yadav wrote:Is xy > 0 ?
(1) x - y > -2
(2) x - 2y < -6

OA is C

[spoiler]It took me 2 mins 47 secs. My first approach was using numbers of different signs and then solving two simultaneous equations in the end. Is there any other faster approach ?

[/spoiler]
Would this be faster? It might be or it might be a source of ideas anyway.

First check Statement 1 by adding y to both sides to create x > y - 2. So if y >= 2, they are both positive. If y < 2, x could be positive, negative or 0. So Statement 1 is insufficient.

Statement 2 similarly becomes x < 2y - 6. If y > 3, then x could be positive, negative or 0. Insufficient.

Then look at the two equations and notice that we can make their left sides the same by adding y to both sides of Statement 2 to get x - y < y - 6.

We can then combine this with Statement 1 to get y - 6 > x - y > -2 So y - 6 > -2 and y > 4.

Substitute 4 into Statement 1 and get x - 4 > -2. x > 2. For any higher y, x is still positive. So all x and all y are positive and xy > 0.

Choose C.
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by GMATGuruNY » Tue Feb 03, 2015 6:57 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.
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by Brent@GMATPrepNow » Tue Feb 03, 2015 10:55 am
sachin_yadav wrote:Is xy > 0 ?

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

Statement 1: x-y > -2
Statement 1 does not FEEL sufficient, so I'm going to test (plug in) some values.
There are several pairs of numbers that meet the condition in statement 1. 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

Aside: For more on this idea of plugging in values when a statement doesn't feel sufficient, you can read my article: https://www.gmatprepnow.com/articles/dat ... lug-values

Statement 2: x - 2y < -6
Statement 2 does not FEEL sufficient either, so let's test (plug in) some values.
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

Here's a similar question: https://www.beatthegmat.com/are-x-and-y- ... 81846.html
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by [email protected] » Tue Feb 03, 2015 6:35 pm
Hi sachin_yadav,

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)

Since 2Y - 6 > Y - 2
2Y - Y > -2 + 6

Y MUST be > 4

Since Y > 4...

X + 2 > 4

X MUST be > 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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Last edited by [email protected] on Mon Nov 07, 2016 10:54 pm, edited 1 time in total.
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by jiamintoh » Wed Feb 04, 2015 12:18 am
I expressed both inequalities in terms of lines.

i.e. From (1),

x - y > -2
x > y - 2
y < x + 2

Charting the line, you will see that this inequality is satisfied in all four cartesian quadrants:

Quadrant I: x is +ve, y is +ve
Quadrant II: x is -ve, y is +ve
Quadrant III: x is -ve, y is -ve
Quadrant IV: x is +ve, y is -ve

Thus, (1) is insufficient. Scratch answers A and D.

From (2),

x - 2y < -6
2y - 6 > x
2y > x + 6
y > x/2 + 3

Charting the line, you will see that this inequality is satisfied in three quadrants:

Quadrant I: x is +ve, y is +ve
Quadrant II: x is -ve, y is +ve
Quadrant III: x is -ve, y is -ve

Therefore, (2) is insufficient, scratch answer B.

Combining both charted lines. You will see that the only area that satisfies both inequalities is in Quadrant I, hence both x and y are both +ve, meaning xy > 0 always.

*Note: for xy > 0, x and y must both be in either the first quadrant (where both are positive), or the third quadrant (where both are negative).

Answer is C. 100 seconds.
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by Matt@VeritasPrep » Mon Feb 09, 2015 1:51 am
Here's one more spiffy way to evaluate the two statements.

Once we've eliminated S1 and S2 alone, we can write the two inequalities as

x > y - 2
2y - 6 > x

Combining the two inequalities, we get 2y - 6 > x > y - 2. Since 2y - 6 > y - 2, we find y > 4. Since x > y - 2, we know x > 4 - 2, or x > 2. So they're both positive, and we're set!
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by Mo2men » Mon Nov 07, 2016 3:59 pm
[email protected] wrote:Hi sachin_yadav,

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)

Since 2Y - 6 > Y - 2
2Y - Y > -2 + 6

Y MUST be > 4

Since Y > 4...

X - 2 > 4

X MUST be > 2

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

Final Answer: C

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

Please review the highlighted part in red as it must be 'x+2>4'NOT 'x-2>4'.

Thanks
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by melguy » Mon Nov 07, 2016 6:18 pm
I think that was already super fast! I attempted this question in 4 mins :oops:
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by Jeff@TargetTestPrep » Fri Nov 11, 2016 7:11 am
sachin_yadav wrote:Is xy > 0 ?
(1) x - y > -2
(2) x - 2y < -6

OA is C
We need to determine whether the product of x and y is positive. We should recall that the product of two numbers is positive only if both the numbers are positive or if both are negative.

Statement One Alone:

x - y > -2

Statement one tells us that the difference between x and y is -2; it does not tell us anything about the signs of x and y. For instance, if x = 2 and y = 1, we have x - y = 1 > -2, and xy is positive. However, if x = 3 and y = -2, 3 - (-2) = 5 > -2, but xy is negative. Statement one alone is not sufficient. We can eliminate answer choices A and D.

Statement Two Alone:

x - 2y < -6

Again, we have a statement that tells us nothing about the signs of x and y. For instance, if x = 3 and y = 5, then x - 2y = 3 - 2(5) = 3 - 10 = -7 < -6, and xy is positive. However, if x = -1 and y = 5, then x - 2y = -1 - 2(5) = -11 < -6, and xy is negative. Statement two alone is not sufficient. We can eliminate answer choice B.

Statements One and Two Together:

Let's manipulate the first inequality to read: y < x + 2. Similarly, we can manipulate the second inequality to read: y > (1/2)x + 3.

Thus, we can say the following:

(1/2)x + 3 < y < x + 2

(1/2)x + 3 < x + 2

x + 6 < 2x + 4

2 < x

Thus, x is positive.

We also know the following:

y > (1/2)x + 3

Since x is greater than two, let's see what we can determine about y, if we substitute 2 for x.

y > (1/2)(2) + 3

y > 4

So y is positive as well. Both statements together are sufficient.

Answer: C

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