Is xy > 0

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GMAT Hacker: Senior | Next Rank: 100 Posts; Posts: 48; Joined: Sun May 30, 2010 12:35 am; Thanked: 1 times

by GMAT Hacker » Tue Aug 24, 2010 6:15 am

Is xy > 0 ?

(1) x - y > -2

(2) x - 2y < -6

I will post the QA later.

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Source: Beat The GMAT — Data Sufficiency | Tue Aug 24, 2010 6:15 am

kmittal82: Master | Next Rank: 500 Posts; Posts: 324; Joined: Mon Jul 05, 2010 6:44 amFollowed by: 3 members; Location: London; Thanked: 70 times

by kmittal82 » Tue Aug 24, 2010 6:52 am

(1)

x - y > -2

=> x > y - 2

Various values of x and y satisfy the above, some yielding xy as +ve, some as -ve.. Not sufficient

(2)

x - 2y < -6

=> 2y - 6 > x

Various values of x and y satisfy this, some yielding xy as +ve, some as -ve. Not Sufficient

Combining:

x - y > -2

=> x - 2y > -2 -y

From (2), we know

x - 2y < -6

Using both, we can say:

-2 -y < -6 => y >4 ( using the logic: b>a, b < c => c > a)

This in turn implies that x has to positive to satisfy x - y > -2 , thus xy will be positive

Hence (C)

Can you confirm the OA?

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gmaterial: Newbie | Next Rank: 10 Posts; Posts: 3; Joined: Tue Mar 27, 2012 10:27 pm

by gmaterial » Tue Apr 03, 2012 12:29 pm

The OA is C.

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Sat Apr 07, 2012 1:01 am

One way to combine the two statements is to link together the inequalities. I posted an explanation here:

https://www.beatthegmat.com/ds-inequalit ... 82218.html

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Shalabh's Quants: Master | Next Rank: 500 Posts; Posts: 134; Joined: Fri Apr 06, 2012 3:11 amFollowed by: 5 members; Thanked: 35 times

by Shalabh's Quants » Sat Apr 07, 2012 5:51 am

Hi,

One way could be...

By Stat.1, x-y>-2 => by dividing both sides by -2. => x/-2-y/-2<1 => x/-2+ y/2<1,

If we plot above linear inequality in XY plane. Line passes through (-2,0) & (0,2) points. It means the line passes through 3 quadrants namely 1,2 & 3.

In quadrants 1; The co-ordinate of x & y will be both +ive.
In quadrants 2; The co-ordinate of x & y will be x -ive & y +ive.
In quadrants 3; The co-ordinate of x & y will be both -ive.

so, product of xy would be +ive in quadrants 1&3, or xy>1.
but,product of xy would be -ive in quadrants 2, or xy<1.

hence stat. 1 is insufficient.

Taking stat. 2...

x - 2y < -6
=> by dividing both sides by -6. => x/-6-2y/-6>1
or, x/-6+y/3>1

If we plot above linear inequality in XY plane. Line passes through (-6,0) & (0,3) points. It means this line too passes through 3 quadrants namely 1,2 & 3.

In quadrants 1; The co-ordinate of x & y will be both +ive.
In quadrants 2; The co-ordinate of x & y will be x -ive & y +ive.
In quadrants 3; The co-ordinate of x & y will be both -ive.

so, product of xy would be +ive in quadrants 1&3, or xy>1.
but,product of xy would be -ive in quadrants 2, or xy<1.

hence stat. 2 is insufficient.

Combining both statements... Simply solve 2 inequalities treating them as linear equation.

x-y>-2 & x-2y<-6;

=> x-y>-2
-x+2y>6;(Changing the sign of inequality)

Which yields y>4 & x>2. or xy>1.

hence stat. 1 & 2 together are sufficient.

Shalabh Jain,
e-GMAT Instructor

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pims: Junior | Next Rank: 30 Posts; Posts: 10; Joined: Tue Aug 16, 2011 7:32 am

by pims » Mon Jul 21, 2014 6:48 am

Rich ?

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by Brent@GMATPrepNow » Mon Jul 21, 2014 7:04 am

GMAT Hacker wrote:Is xy > 0 ?

(1) x - y > -2

(2) x - 2y < -6

I will post the QA later.

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 > 0
Case b: x = 1 and y = -1, in which case 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 = -10 and y = -1, in which case xy > 0
Case b: x = -10 and y = 1, in which case xy < 0
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
NOTE: We can ADD inequalities if the inequality signs are facing the SAME DIRECTION.
Statement 1 says x - y > -2
Take statement 2 and multiply both sides by -1 to get: -x + 2y > 6
Now that the inequality signs are facing the SAME DIRECTION, add the inequalities to get:
y > 4
In other words, y is POSITIVE
Also, since y > 4 and x > y-2, we know that x > 2. In other words, x is POSITIVE
If x and y are both POSITIVE, then 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

Brent Hanneson - Creator of GMATPrepNow.com

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GMATinsight: Legendary Member; Posts: 1100; Joined: Sat May 10, 2014 11:34 pmFollowed by: 24 members; Location: New Delhi, India; Thanked: 205 times

by GMATinsight » Mon Jul 21, 2014 8:03 am

GMAT Hacker wrote:Is xy > 0 ?

(1) x - y > -2

(2) x - 2y < -6

I will post the QA later.

Question : Is XY > 0?
but if X and Y have same Sign then Answer to the question will be YES
and if X and Y have Different Sign then Answer to the question will be NO

Therefore we only have to find out whether their signs are same or not

Question Rephrased : Do X and Y have same Sign?

Statement 1) X-Y > -2
which can be true for X=2 and Y = -1 (X and Y different sign) and also if
which can be true for X=2 and Y = 1 (X and Y same sign)

Inconsistent Solution therefore INSUFFICIENT

Statement 2) X-2Y < -6
which can be true for X=-10 and Y = 1 (X and Y different sign) and also if
which can be true for X=2 and Y = 10 (X and Y same sign)

Inconsistent Solution therefore INSUFFICIENT

Combining the two statements
X-2Y < -6 i.e. -X+2Y >6
and
X-Y > -2

Adding the two equations we get
(-X+2Y)+(X-Y) > 6-2
i.e. Y > 4
i.e. Y is positive
from Statement 1
X-Y > -2
i.e. X > -2+(4)
i.e. X > 2
i.e. X is positive

Both are positive therefore

SUFFICIENT

Answer: Option C

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by [email protected] » Mon Jul 21, 2014 1:58 pm

Hi pims,

This is a rather "lean" DS question - it has no restrictions; it's just a question. Is XY > 0? This is a YES/NO question.

This is a great question for TESTing VALUES. In these situations, I love using the number 0 (if I'm not restricted from using it by something in the Facts). This question is also built around some Number Properties:

For XY to be greater than 0, then....
X and Y are BOTH positive OR BOTH negative (which gives us a YES answer)

ANY other combination (1 positive and 1 negative, anything and a 0) will give you an XY that is NOT greater than 0 (which gives us a NO answer).

Fact 1: X - Y > -2

Let's TEST VALUES
If....
X = 1, Y = 0, then the answer to the question is NO
X = 1, Y = 1, then the answer to the question is YES
Fact 1 is INSUFFICIENT

Fact 2: X - 2Y < -6

Again, let's TEST VALUES
If...
X = -10, Y = 0, then the answer to the question is NO
X = -10, Y = -1, then the answer to the question is YES
Fact 2 is INSUFFICIENT

Combined, we know...
X - Y > -2
X - 2Y < -6

From here, we can "do math" in a variety of ways. I'm going to rewrite the inequalities....

X > Y - 2
X < 2Y - 6

And then combine them...

2Y - 6 > X > Y - 2

We now know....

2Y - 6 > Y - 2

Simplify...

Y > 4

Plugging that information back into this inequality...

X > Y - 2

X > (>4) - 2

....means that X MUST be POSITIVE.

So, Y is POSITIVE and X is POSITIVE, so the answer to the question is always YES.
Combined, SUFFICIENT.

Final Answer:C

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

Contact Rich at [email protected]

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