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# Is |x-y|>|x|-|y|

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Is |x-y|>|x|-|y| ? [#permalink]  07 May 2012, 08:08
OA is B. I faced this problem in the GMATPrep.

(1) y < x
if x=3 and y=2, left hand abs(x-y) = 1, and right hand abs(x) - abs(y) = 1...No.
But if x=3 and y=-2, left hand is 5 and right is 1...Yes.
INSUFFICIENT.

(2) xy < 0
Let's think the following two cases.

(a) x>0 and y<0
In this case abs(x-y) > abs(x), as in the second plug-in in the discussion of (1) above.
So abs(x-y) naturally is greater than abs(x) - abs(y) because abs(x) > abs(x)-abs(y)...Yes.

(b) x<0 and y>0
In this case abs(x-y) = abs(x)+abs(y). Since abs(x) + abs(y) > abs (x) - abs(y),
abs(x-y) > abs(x)-abs(y)...Yes.

SUFFICIENT.

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Re: Is |x - y | > |x | - |y | ? (1) y < x (2) xy < 0 [#permalink]  06 Mar 2013, 21:36

1) Taking statement (1), if y < x, then there can be two cases -
a) y is negative, it can lead to two sub cases --
(i) x is negative ==> as y < x so |y| > |x| ==> |x| - |y| will be < 0, and |x - y| > 0 ==> |x - y| > |x| - |y|
(ii) x is positive ==> |x - y| would be sum of absolute value of x and y, essentially |x| + |y| ==> |x - y| > |x| - |y|

Problem statement is true.

b) y is positive ==> x can only be positive ==> |x - y| = |x| - |y|

Problem statement is false.

Since we do not know, whether y is positive or negative we can not conclude from statement 1.

2) Taking statement (2), if xy < 0 ==> two sub cases
a) x < 0 and y > 0 ==> |x - y| = |x| + |y| which is greater than |x| - |y|
b) x > 0 and y < 0 ==> |x - y| = |x| + |y|, which is again greater than |x| - |y|

Statement (2) is sufficient enough to conclude the problem statement.

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Re: Is |x-y|>|x|-|y| [#permalink]  08 Mar 2013, 04:56
So basically the explanation is that find the signs test in the following cases

++
- -
+ -
- +

since statement doesn't 1 doesn't help in any way its insufficient and statement 2 either both positive or both negative when we plug examples its never true so its sufficient?
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Re: Is |x-y|>|x|-|y| [#permalink]  25 Mar 2013, 02:08
mmphf wrote:
Is |x-y|>|x|-|y| ?

(1) y < x
(2) xy < 0

How it is B? Did they mention that X and Y are integers? No right, the answer should be E. If they provide details about X and Y as integers then it will be B otherwise it will be E.

can anyone help me about the scenario whether we consider fractions or not in this case?

Scenario:

x=1/2, y=1/3 ==> |1/2-1/3|=1/6 and |1/2|-|1/3|=1/6
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Re: Is |x-y|>|x|-|y| [#permalink]  25 Mar 2013, 03:56
Expert's post
kancharana wrote:
mmphf wrote:
Is |x-y|>|x|-|y| ?

(1) y < x
(2) xy < 0

How it is B? Did they mention that X and Y are integers? No right, the answer should be E. If they provide details about X and Y as integers then it will be B otherwise it will be E.

can anyone help me about the scenario whether we consider fractions or not in this case?

Scenario:

x=1/2, y=1/3 ==> |1/2-1/3|=1/6 and |1/2|-|1/3|=1/6

The point is that x = 1/2 and y = 1/3 do not satisfy xy < 0 (the second statement).
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Re: Is |x-y|>|x|-|y| [#permalink]  25 Mar 2013, 04:37
Bunuel wrote:
kancharana wrote:
mmphf wrote:
Is |x-y|>|x|-|y| ?

(1) y < x
(2) xy < 0

How it is B? Did they mention that X and Y are integers? No right, the answer should be E. If they provide details about X and Y as integers then it will be B otherwise it will be E.

can anyone help me about the scenario whether we consider fractions or not in this case?

Scenario:

x=1/2, y=1/3 ==> |1/2-1/3|=1/6 and |1/2|-|1/3|=1/6

The point is that x = 1/2 and y = 1/3 do not satisfy xy < 0 (the second statement).

Thank you... understood that.
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Re: Is |x-y|>|x|-|y| [#permalink]  25 Mar 2013, 10:19
kancharana wrote:
mmphf wrote:
Is |x-y|>|x|-|y| ?

(1) y < x
(2) xy < 0

How it is B? Did they mention that X and Y are integers? No right, the answer should be E. If they provide details about X and Y as integers then it will be B otherwise it will be E.

can anyone help me about the scenario whether we consider fractions or not in this case?

Scenario:

x=1/2, y=1/3 ==> |1/2-1/3|=1/6 and |1/2|-|1/3|=1/6

It's implied that it is integers on the GMAT? Is this perception by me correct or completely out of the blue?
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Re: Is |x-y|>|x|-|y| [#permalink]  25 Mar 2013, 11:39
Expert's post
kancharana wrote:
mmphf wrote:
Is |x-y|>|x|-|y| ?

(1) y < x
(2) xy < 0

How it is B? Did they mention that X and Y are integers? No right, the answer should be E. If they provide details about X and Y as integers then it will be B otherwise it will be E.

can anyone help me about the scenario whether we consider fractions or not in this case?

Scenario:

x=1/2, y=1/3 ==> |1/2-1/3|=1/6 and |1/2|-|1/3|=1/6

From F.S 1, we have that x>y. Thus |x-y| = x-y. Thus, we have to answer whether x-y>|x|-|y|.

or x-|x|>y-|y|. Now for x>0, and y>0, we have is 0>0 and hence a NO. Again, for x>0 and y<0, we have a YES. Insufficient.

For F.S 2, we know that x and y are of opposite signs. Thus, x and y being on the opposite sides of the number line w.r.t the origin, the term |x-y| will always be more than the difference of the absolute distance of x and y from origin.Sufficient.

B.
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Last edited by mau5 on 05 Apr 2013, 04:02, edited 1 time in total.
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Re: Is |x-y|>|x|-|y| [#permalink]  26 Mar 2013, 01:06
Expert's post
tulsa wrote:
kancharana wrote:
mmphf wrote:
Is |x-y|>|x|-|y| ?

(1) y < x
(2) xy < 0

How it is B? Did they mention that X and Y are integers? No right, the answer should be E. If they provide details about X and Y as integers then it will be B otherwise it will be E.

can anyone help me about the scenario whether we consider fractions or not in this case?

Scenario:

x=1/2, y=1/3 ==> |1/2-1/3|=1/6 and |1/2|-|1/3|=1/6

It's implied that it is integers on the GMAT? Is this perception by me correct or completely out of the blue?

No, that's completely wrong, we cannot assume that x and y are integers, if this is not explicitly stated.

Generally, GMAT deals with only Real Numbers: Integers, Fractions and Irrational Numbers. So, if no limitations, then all we can say about a variable in a question that it's a real number.

For more check here: math-number-theory-88376.html

Hope it helps.
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Re: Is |x-y|>|x|-|y| [#permalink]  04 Apr 2013, 11:18
Bunuel wrote:
mmphf wrote:
Is |x-y|>|x|-|y| ?

(1) y < x
(2) xy < 0

Is |x-y|>|x|-|y|?

Probably the best way to solve this problem is plug-in method. Though there are two properties worth to remember:
1. Always true: $$|x+y|\leq{|x|+|y|}$$, note that "=" sign holds for $$xy\geq{0}$$ (or simply when $$x$$ and $$y$$ have the same sign);

2. Always true: $$|x-y|\geq{|x|-|y|}$$, note that "=" sign holds for $$xy>{0}$$ (so when $$x$$ and $$y$$ have the same sign) and $$|x|>|y|$$ (simultaneously). (Our case)

So, the question basically asks whether we can exclude "=" scenario from the second property.

(1) y < x --> we can not determine the signs of $$x$$ and $$y$$. Not sufficient.
(2) xy < 0 --> "=" scenario is excluded from the second property, thus $$|x-y|>|x|-|y|$$. Sufficient.

(1) x>y
x=-2,y=-4 then 2>-2 --> yes
x=4,y=-2 then 6>2 --> yes
can't get a no, so sufficient

(2) xy<0
x=4,y=-2 then 6>2 --> yes
x=-2,y=4 then 6>-2 --> yes
can't get a no, so sufficient

ans: D
why is the answer B? is the question mis-written and the inequality sign should have >= or <=?
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Re: Is |x-y|>|x|-|y| [#permalink]  05 Apr 2013, 01:55
Expert's post
margaretgmat wrote:
Bunuel wrote:
mmphf wrote:
Is |x-y|>|x|-|y| ?

(1) y < x
(2) xy < 0

Is |x-y|>|x|-|y|?

Probably the best way to solve this problem is plug-in method. Though there are two properties worth to remember:
1. Always true: $$|x+y|\leq{|x|+|y|}$$, note that "=" sign holds for $$xy\geq{0}$$ (or simply when $$x$$ and $$y$$ have the same sign);

2. Always true: $$|x-y|\geq{|x|-|y|}$$, note that "=" sign holds for $$xy>{0}$$ (so when $$x$$ and $$y$$ have the same sign) and $$|x|>|y|$$ (simultaneously). (Our case)

So, the question basically asks whether we can exclude "=" scenario from the second property.

(1) y < x --> we can not determine the signs of $$x$$ and $$y$$. Not sufficient.
(2) xy < 0 --> "=" scenario is excluded from the second property, thus $$|x-y|>|x|-|y|$$. Sufficient.

(1) x>y
x=-2,y=-4 then 2>-2 --> yes
x=4,y=-2 then 6>2 --> yes
can't get a no, so sufficient

(2) xy<0
x=4,y=-2 then 6>2 --> yes
x=-2,y=4 then 6>-2 --> yes
can't get a no, so sufficient

ans: D
why is the answer B? is the question mis-written and the inequality sign should have >= or <=?

What about the case x = 4, y = 2 in statement 1?
then we get 2 > 2 --> No
Hence statement 2 is not sufficient.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Senior Manager Joined: 13 May 2013 Posts: 475 Followers: 1 Kudos [?]: 98 [0], given: 134 Re: Is |x-y|>|x|-|y| [#permalink] 26 Jun 2013, 07:17 Is |x-y|>|x|-|y| ? (1) y < x If y is less than x then (x-y) is going to be positive, however, we don't know if x and y are positive or negative: I. (x-y) > x -y ===> 0 > 0 II. (x-y) > -x -y ===> 2x > 0 III. (x-y) > -x +y ===> 2x > 2y IV. (x-y) > x +y ===> 0 > 2y The way I see it, is with case I.) 0>0 isn't true, II.) x must be some non-negative # that isn't zero, III.) x > y but we already know that, IV.) y must be some non-negative # that isn't zero. So we know that x is positive, y is negative and that x > y but we still can't get a single answer. All we know for sure is that y < x (x=-2, y=-4) |x-y|>|x|-|y| (x-y)>(-x)-(-y) x-y>-x+y 2x>2y x>y |-2-(-4)|>|-2|-|-4| |2|>|2|-|4| 2>-2 TRUE (x=2, y=-4) |x-y|>|x|-|y| (x-y)>x-(-y) x-y>x+y 0>2y |x-y|>|x|-|y| |2-(-4)|>|2|-|-4| 6>-2 TRUE (x=4, y=2) |x-y|>|x|-|y| (x-y)>(x)-(y) x-y>x-y 0>0 |x-y|>|x|-|y| |4-2|>|4|-|2| 2>2 FALSE (0>0 isn't possible, nor does it confirm y or x) NOT SUFFICIENT (2) xy < 0 So either x is less than zero or y is less than zero. x & y ≠ 0. There are two possible cases: (x is positive and y is negative) or (x is negative and y is positive) I. (x is positive and y is negative) |x-y|>|x|-|y| (x-y)>(x)-(-y) x-y>x+y 0>2y (which holds with the premise in the first case that y is negative) II. (x is negative and y is positive) |x-y|>|x|-|y| -(x-y)>(-x)-(y) -x+y>-x-y 2y>0 (which holds with the premise in the second case that y is positive) SUFFICIENT (B) (does that make sense?) Senior Manager Joined: 23 Jan 2013 Posts: 423 Schools: Cambridge'16 Followers: 2 Kudos [?]: 33 [0], given: 34 Re: Is |x-y|>|x|-|y| [#permalink] 06 Aug 2013, 08:28 Is |x-y|>|x|-|y| ? (1) y < x (2) xy < 0 what if like that (x-y)^2>x^2-y^2 so x^2-2xy+y^2>x^2-y^2 and x^2-2xy+y^2-x^2+y^2>0, and 2y^2-2xy>0 and 2y(y-x)>0 finally, y>0 and y-x>0 (y>x) Then, 1) y < x, not sufficient, because it negates only one final condition and y may be both positive and negative 2) xy < 0, sufficient, because confirms that when y>0, y>x when x is negative B write, if it is OK Senior Manager Joined: 10 Jul 2013 Posts: 343 Followers: 3 Kudos [?]: 209 [0], given: 102 Re: Is |x-y|>|x|-|y| [#permalink] 06 Aug 2013, 13:29 kancharana wrote: mmphf wrote: Is |x-y|>|x|-|y| ? (1) y < x (2) xy < 0 How it is B? Did they mention that X and Y are integers? No right, the answer should be E. If they provide details about X and Y as integers then it will be B otherwise it will be E. can anyone help me about the scenario whether we consider fractions or not in this case? Scenario: x=1/2, y=1/3 ==> |1/2-1/3|=1/6 and |1/2|-|1/3|=1/6 ....... st(1), use x=3 , y = 2 and then x=1 , y = -1 , we will have a double case. ----insufficient st(2), use x= -5 , y = 10 and then x=10 , y = -5, we will have a single solution and its yes |x-y|>|x|-|y| .so its sufficient. you can use fractions in st(2) maintaining one positive and the other negative. st(2) will provide the same. so Answer is (B) _________________ Asif vai..... Manager Status: Persevering Joined: 15 May 2013 Posts: 225 Location: India Concentration: Technology, Leadership GMAT Date: 08-02-2013 GPA: 3.7 WE: Consulting (Consulting) Followers: 1 Kudos [?]: 67 [0], given: 34 Re: Is |x-y|>|x|-|y| [#permalink] 24 Aug 2013, 05:11 kancharana wrote: mmphf wrote: Is |x-y|>|x|-|y| ? (1) y < x (2) xy < 0 How it is B? Did they mention that X and Y are integers? No right, the answer should be E. If they provide details about X and Y as integers then it will be B otherwise it will be E. can anyone help me about the scenario whether we consider fractions or not in this case? Scenario: x=1/2, y=1/3 ==> |1/2-1/3|=1/6 and |1/2|-|1/3|=1/6 It really does not matter; no one is saying that they are integers. The problem with your approach is that you considered invalid values for the fractions. According to b xy<0; so either x or y must be -ve. Now, lets put the valid values as x=1/2 and y=-1/3; In LHS we get |1/2+1/3|=5/6 and in RHS we get 1/6; therefore the inequality holds, hence statement b is sufficient. _________________ --It's one thing to get defeated, but another to accept it. Intern Joined: 06 Apr 2012 Posts: 10 Followers: 0 Kudos [?]: 1 [0], given: 34 Re: Inequlities [#permalink] 28 Jun 2014, 19:32 First of all we need to consider different cases to solve this problem. take option 1) y<x this option can be subdivided into two blocks...when both are x,y>0 and x>y. lets take x=2, y=1 lx-yl = l2 -1l = 1 Right hand side of the equation = lxl - lyl = l2l - l1l = 1....so equation is invalid. lets take another example when x= 1 and y = -1... lx-yl = l1 - (-1)l = 2 and Rgiht hand side = 0 which make our equation valid....hence we cannot conclude anything from this option. take option 2) xy<0 under this option there can be two cases....a) x>0 and y<0 (b) x<0 and y>0 lets take a) and use some values.... x=2 and y = -1... simplifing the equation we get...lx-yl = 3 where lxl - lyl = 1 it makes equation valid. now take b) x= -2 and y = 1...we get lx-yl = 3 and lxl - lyl = 1 its also satisfy our given equation. so this option is sufficent to answer the given question. OA (B) Intern Joined: 23 Sep 2014 Posts: 15 Followers: 0 Kudos [?]: 0 [0], given: 6 Re: Is |x-y|>|x|-|y| [#permalink] 17 Dec 2014, 04:45 I really never understand these questions in general about absolute values. What exactly is the difference between |x-y| and |x| - |y| ? Math Expert Joined: 02 Sep 2009 Posts: 29776 Followers: 4896 Kudos [?]: 53434 [0], given: 8160 Re: Is |x-y|>|x|-|y| [#permalink] 17 Dec 2014, 04:50 Expert's post JoostGrijsen wrote: I really never understand these questions in general about absolute values. What exactly is the difference between |x-y| and |x| - |y| ? One should go through basics and brush fundamentals first and only after that practice questions, especially hard ones. Theory on Abolute Values: math-absolute-value-modulus-86462.html Absolute value tips: absolute-value-tips-and-hints-175002.html DS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=37 PS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=58 Hard set on Abolute Values: inequality-and-absolute-value-questions-from-my-collection-86939.html _________________ Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 5964 Location: Pune, India Followers: 1527 Kudos [?]: 8414 [0], given: 193 Re: Is |x-y|>|x|-|y| [#permalink] 17 Dec 2014, 07:08 Expert's post JoostGrijsen wrote: I really never understand these questions in general about absolute values. What exactly is the difference between |x-y| and |x| - |y| ? Try to put in values for x and y (positive as well as negative) and that will help you see the difference between these expressions. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: Is |x-y|>|x|-|y| [#permalink]  11 Jan 2015, 05:48
I have recently started with the prep thus looking for patterns to solve the Absolute Value questions.
Wanted to check if the below solution stands valid.

Square both the sides of the question stem
|x-y|^2>(|x|-|y|)^2 --- |x-y|^2 = (x-y)^2
x^2+y^2-2xy>x^2+y^2-2|x||y|
xy<|x||y|

From 1
y<x - In Sufficient

From 2
xy<0 (Either of them is negative i.e. x +ve y -ve or y +ve or x -ve)
Sufficient
Re: Is |x-y|>|x|-|y|   [#permalink] 11 Jan 2015, 05:48

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