Is xy 0? (1) x - y -2 (2) x - 2y

Question

Is xy > 0?

(1) x - y > -2
(2) x - 2y < -6

Bunuel · Answer

How are you getting 12 - x > 10? 3y - x > 10 3y > 12. We cannot subtract those two the way you did. You can only apply subtraction when their signs are in the opposite directions : If a>b and c and < ) --> a-c>b-d (take the sign of the inequality you subtract from). Example: 3<4 and 5>1 --> 3-5<4-1 . You can only add inequalities when their signs are in the same direction: If a>b and c>d (signs in same direction: > and > ) --> a+c>b+d . Example: 3<4 and 2<5 --> 3+2<4+5 . For more check Manipulating Inequalities .

LauraOrion · Answer

This question is fantastic for bringing up a strategy not enough people think to do with inequalities: as long as you have the inequality signs pointed in the same direction, you can add the inequalities together to eliminate a variable the same way you would for a system of equations.

Here once you've eliminated A, B, and D by quick conceptual number picking**, you can multiply the Statement 2 inequality by -1 so that you have both inequalities with signs in the same direction:

x - y > -2

and

2y - x > 6

If you then add the two inequalities, you eliminate the x term (one is positive and the other is negative), leaving:

y > 4

So you know y is positive, and then when you plug y > 4 in to the first equation (which can be expressed as x > y - 2), you know that x is greater than "something greater than 4, minus 2" so x must also be positive, so you know that the product xy is greater than 0. Together the statements are sufficient and the answer is (C).

Note that when you do have two inequalities this situation presents itself more often than most people think! If you can get the signs facing in the same direction (generally by multiplying one inequality by -1 if they're not already in the same direction) you can add the inequalities together (don't subtract, since "minus" is basically "plus negative" and can screw up the positive/negative inequality logic) and solve like a system of equations.

** Try x = 4 and y = 6 versus x = -2 and y = 0 for statement 1 and x= 0 and y = 3 and x = 1 and y = 12 for statement 2, for example.

chetan2u · Answer

Is xy > 0 ?

This basically asks us .."Are x and y of SAME sign?"

(1) x – y > –2
x>y-2..
y=2, x=4...yes
y=-2, x=4...no
Insufficient

(2) x – 2y < –6
x<2y-6..
y=2, x=-4...no
y=-2, x=-15...yes
Insufficient

Combined
x-y>-2
x-2y<-6 or 2y-x>6
Add the two inequalities
x-y+2y-x>-2+6.......y>4
Multiply first by 2 and add the equations to get x
2(x-y)+2y-x>2*(-2)+6.....2x-2y+2y-x>6-4.....x>2

So both x and y are positive
Ans is always yes
Sufficient

C

BrentGMATPrepNow · Answer

Target question : Is xy > 0? Statement 1 : x - y > -2 Let's TEST some values. There are several values of x and y that satisfy statement 1. Here are two: Case a : x = 5 and y = 1. In this case, xy = (5)(1) = 5. So, the answer to the target question is YES, xy IS greater than 0 Case b : x = 5 and y = -1. In this case, xy = (5)(-1) = -5. So, the answer to the target question is NO, 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 Let's TEST some values again. There are several values of x and y that satisfy statement 2. Here are two: Case a : x = 1 and y = 10. In this case, xy = (1)(10) = 10. So, the answer to the target question is YES, xy IS greater than 0 Case b : x = -1 and y = 10. In this case, xy = (-1)(10) = -10. So, the answer to the target question is NO, 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 IMPORTANT: There’s a useful inequality property that says If two inequalities have their inequality symbols facing the SAME DIRECTION, we can ADD the inequalities. We have: x - y > -2 x - 2y < -6 Take the BOTTOM equation and multiply both sides by -1 to get: x - y > -2 -x + 2y > 6 Now ADD the equations to get: y > 4 In other words, y is POSITIVE Now take: x - y > -2 -x + 2y > 6 Take the TOP equation and multiply both sides by 2 to get: 2x - 2y > -4 -x + 2y > 6 Now ADD the equations to get: x > 2 In other words, x is POSITIVE Since x and y are both POSITIVE, we know that xy IS positive Since we can answer the target question with certainty, the combined statements are SUFFICIENT Answer: C RELATED VIDEOS https://www.youtube.com/watch?v=Od98BOicIq4 https://www.youtube.com/watch?v=S_fjYmCtvy0

Kinshook · Answer

Is xy>0 1) x-y > -2 x>y-2 xy>y(y-2)=y^2-2y Value of y is unknown NOT SUFFICIENT 2) x-2y < -6 x<2y-6 xy -2 x>y-2. (3) 2) x-2y < -6 x<2y-6 -x>-2y+6 (4) Adding (3) & (4) 0>-y+4 y>4>0 (5) From equation (3) x>y-2>4-2=2>0 x>2>0 (6) xy>0 SUFFICIENT IMO C

avigutman · Answer

Video solution from Quant Reasoning (starts at 0:00:25): Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1 https://www.youtube.com/watch?v=cn-ojY-2Gj0

UserMaple5 · Answer

EgmatQuantExpert   Bunuel   JeffTargetTestPrep

Hi experts. I tend to get these types of questions right but they take me WAY TOO LONG to do. I normally test cases and it takes a while. Are there any strategies other than testing cases that I can use for these kind of questions to help improve timing?

Thank you in advance!

GMATinsight · Answer

Question; Is xy > 0? STatement 1 : x - y > -2 @x = 1, y may be +1 i.e. xy > 0 @x = 1, y may be -1 i.e. xy < 0 NOT SUFFICIENT STatement 2 :x - 2y < -6 @x = 1, y may be +4 i.e. xy > 0 @x = -1, y may be +4 i.e. xy < 0 NOT SUFFICIENT Combining the statements Rule of Inequalities : - Two inequalities may be added if their inequality signs are identical - Two inequalities may be Subtracted if their inequality signs are Opposite Rule 1 is enough

Now we have x - y > -2 and x - 2y < -6 ie.. x - y > -2 and -x + 2y > 6 (multiplied -1 both sides in second inequation) Adding the two inequations x - y -x + 2y > -2+6 i.e. y > 4 Substituting it in inequation 1 x - y > -2 i.e. x > -2+y i.e. x > 2 SInce both x and y are positive, therefore xy > 0 Answer: Option C

avigutman · Answer

UserMaple5 yes, you can use number line reasoning to avoid having to test cases. You can see how I do that in the first part of the video from this post: https://gmatclub.com/forum/is-xy-0-1-x- ... l#p2686547

CrackverbalGMAT · Answer

This is a question which is a great example of what the GMAT expects from test takers in terms of combining statements in a DS question on Inequalities.

It is also an excellent question to learn how to prove/disprove the individual statements in a DS question.

Breaking down the question stem, we can rephrase the question as “Are x and y of the same signs?”. The answers to this question could be,

Yes – which means x and y are either both positive or both negative
No – which means one of them could be positive and the other could be negative OR ZERO (many of us ignore ZERO at our own peril).

From statement I alone, x-y > -2. Let us take some values to find out if this information is sufficient to answer the main question.
If x = 10 and y = 5, x-y = 5 which is definitely greater than -2. For these values of x and y, we answer the main question with a Yes.
If x = 10 and y = -5, x-y = 15 which is also greater than -2. For these values of x and y, we answer the main question with a No.

Statement I alone is insufficient to answer the question with a definite YES or NO. Answer options A and D can be eliminated. Possible answer options are B, C or E.

From statement II alone, x-2y < -6. Again, as in the first statement, let us test some values.
If x = 5, y = 10, x – 2y = -15 which is less than -6. These values give a YES to the main question.
If x = -5, y = 10, x – 2y = -25 which is less than -6. These values give a NO to the main question.

Statement II alone is insufficient to answer the question with a definite YES or NO. Answer options B can be eliminated. Possible answer options are C or E.

Now, at this stage, we are required to combine the statements I and II. Here’s where GMAT expects you to combine the statements using an important property of inequalities and that is – Two inequalities can be added if they have the same inequality sign.

The two inequalities are: x – y > -2 and x – 2y < -6.

When we multiply the first inequality with -2, we obtain -2x + 2y < 4. Now, since the inequality sign is the same as the second inequality, we can add these inequalities.
Adding the inequalities, 2y and -2y cancel out each other and we have –x < -2 which can be rewritten as x > 2.

When we multiply the second inequality with -1, we obtain –x + 2y > 6. Since the inequality sign is the same as the first, we can add these inequalities.
Adding them, x and –x cancel out each other and we have y > 4.

This means xy > 0 and we can answer the main question with a Yes. The combination of statements is sufficient, answer option E can be eliminated.

The correct answer option is E.

Remember that inequalities can be added as long as they have the same sign.

Hope that helps!
Aravind B T

JuliaMaria123 · Answer

I think it should be possible to skip all the calculations and graphical solutions and number line drawings and just think about this for a few moments.

So it is clear by just picking a few numbers that neither (1) nor (2) are sufficient by itself.

So now the big question is - are they sufficient together?

We have: x - y > -2 and x - 2y < -6 from statement (1) and (2).

We can rewrite statement (2): x - y - y < -6 and the red-coloured part much be larger than -2, as it is given by statement (1). But statement (2) must be smaller than -6. We can only achieve that if y is positive (notice the negative sign) and must be great than 4 to make statement (2) smaller than -6).

So now we know that y is positive and greater than 4 and we can go back to (1): x - y > -2. If y is greater than 4, we now know that x must be positive so that (1) is great than -2.

GMATGuruNY · Answer

Do x and y have the same sign?

Statement 1, rephrased: x > y-2  
Clearly, x and y can have the same sign or different signs.
INSUFFICIENT

Statement 2, rephrased: x < 2y-6 
Clearly, x and y can have the same sign or different signs.
INSUFFICIENT.

Inequalities with multiple variables can be ADDED TOGETHER.
Before adding, we must ensure that the inequality symbol faces the same direction in each inequality.

Statements combined: 
Adding together x > y-2 and 2y-6 > x, we get:
x+2y-6 > y-2+x
y > 4

Adding together y > 4 and x > y-2, we get:
y+x > 4+y-2
x > 2

Since x>2 and y>4, x and y have the same sign.
Thus, the answer to the question stem is YES.
SUFFICIENT.

C

Bambi2021 · Answer

Taken together:

Subtracting inequalities of different signs -->

(x-y)-(x-2y) > (-2)-(-6)

y > 4

From s1 it is now obvious that x must also be positive.

Posted from my mobile device

forrever · Answer

Hey Bunuel!
I am a bit slow when it comes to thinking of numbers to solve DS Questions like these. I am quite strong in visualising the graphs when it comes to equations. So I started by plotting both the lines after quickly ruling out answer choices (A) and (B). After plotting, I selected the common area i.e the intersection of x - y > -2 and x - 2y < -6 and the common area of these two equations lies completely in the first quadrant, therefore, both x and y will always be positive and we can conclude that (C) is the correct answer. Is there any downside to this approach? I know all the graphs of the basic equations that I have seen on 750, or 700 level questions, what other graphs should I know about if I want to solve problems like these using the graph approach?

Engineer1 · Answer

Bunuel   KarishmaB   MartyMurray 
Generic question:
I did get it right in my practice exam (1:34) and I plugged in numbers as couple of the solutions show in this discussion. But I typically do not feel confident mostly because I don't want to keep spending time to cross check / revise. What should I do? What is a recommended method to save time and also inadvertently cross check solution for inequality question types? I don't think I have "problem" with the concept and if I have more time, I do these questions well. Thank you.

KarishmaB · Answer

The graphical solution Bunuel has given above is a 1 min solution and that would be my method of choice. 
Plugging in numbers is generally not a good strategy for DS questions. At most, it can help you arrive at the pattern in a question.

vibhasharma · Answer

Hi  Bunuel ,

Could you please help with this? Are we missing something here?

vibhasharma · Answer

Hi  Bunuel ,

Could you please help with this? Are we missing something here?

Bunuel · Answer

That doubt has already been addressed here: https://gmatclub.com/forum/is-xy-0-1-x- ... l#p1989838

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

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