Is xy > 0 ?
(1) x - y > -2
(2) x - 2y < -6
I will post the QA later.
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Is xy > 0
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(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?
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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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
https://www.beatthegmat.com/ds-inequalit ... 82218.html
Last edited by GMATGuruNY on Mon Jul 21, 2014 8:12 am, edited 1 time in total.
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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.
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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Target question: Is xy > 0 ?GMAT Hacker wrote:Is xy > 0 ?
(1) x - y > -2
(2) x - 2y < -6
I will post the QA later.
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
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Question : Is XY > 0?GMAT Hacker wrote:Is xy > 0 ?
(1) x - y > -2
(2) x - 2y < -6
I will post the QA later.
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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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
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
