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

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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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New post 02 Jan 2014, 22:36
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study wrote:
Is |x - y| > |x| - |y|?

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


Statement I is insufficient

x....y.....(|x|-|y|)....|x-y|.....YES/NO
3....2.......1..............1........NO
3..-3........0..............6........YES

Statement II is sufficient
xy<0 means that either one of the numbers is negative and other is positive

|x-y| will always result to be a greater value as negative - positive or positive - negative will result in addition of the numbers and |x| - |y| will always result in subtraction.

Hence the answer is B
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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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New post 03 Jan 2014, 03:40
rawjetraw wrote:
Bunuel wrote:
SujD wrote:

Bunuel,
for 1.B when .. y ..0 .. x, you said \(|x-y|>|x|-|y|\) becomes: \(x-y>x+y\).
and
1.c when ... y ... x ... 0, you said \(|x-y|>|x|-|y|\) becomes: \(x-y>-x+y\) --> \(x>y\).

Can you explain this a little bit more? How did you go about removing the absolute signs for this scenarios?

Thanks for your help.



Consider absolute value of some expression - \(|some \ expression|\):
If the expression in absolute value sign (||) is negative or if \(some \ expression<0\) then \(|some \ expression|=-(some \ expression)\);
If the expression in absolute value sign (||) is positive or if \(some \ expression>0\) then \(|some \ expression|=some \ expression\).

(It's the same as for \(|x|\): if \(x<0\), then \(|x|=-x\) and if \(x>0\), then \(|x|=x\))

We have \(|x-y|>|x|-|y|\):

For B: ---------\(y\)---\(0\)---------\(x\)---, \(y<0<x\) (\(x>y\)) --> so as \(x-y>0\), then \(|x-y|=x-y\). Also as \(x>0\), then \(|x|=x\) and as \(y<0\), then \(|y|=-y\). So in this case \(|x-y|>|x|-|y|\) becomes: \(x-y>x-(-y)\) or \(x-y>x+y\) --> \(2y<0\) --> \(y<0\). Which is right, as we consider the range \(y<0<x\);

The same for C.

Hope it's clear.



Hi Bunuel,
Thanks for this great explanation. But I am still unclear about this.

you said... for the case B.. \(y<0<x\) (\(x>y\)) --> so as \(x-y>0\), then \(|x-y|=x-y\). I understand this.

but how is this the same when it comes to case C. where x and y both are negative (y<x<0), although i do get that X is still greater than Y but I am confused how would it still translate to \(|x-y|=x-y\) when both of them are negative... wouldnt it be more like \(|x-y|=y-x\) , i get the RHS part ... its the LHS where I am confused. :roll: :?:

please explain
thanks!


Consider x=-1 and y=-2 for C: in this case \(|x-y|=|-1-(-2)|=1=x-y\).

Hope it helps.
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Squaring and simplifying, is xy < |x||y|?
1. x > y. put x = -2, y = -4 NO; put x = 4, y = -2 YES. NOT SUFFICIENT
2. xy <0; |x||y| always > 0, so xy always < |x||y| SUFFICIENT

B.

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New post 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)

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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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Is this a valid approach to solve this problem?

| X –Y | > |X| - |Y|
Squaring both sides
(X-Y)^2>(|X|-|Y|)^2

X^2-2XY+Y^2>X^2-2|XY|+Y^2

-XY>|XY|

XY<|XY| --> Can be true only for XY < 0.

1 : y > X - Insufficient
2 : XY < 0 -> Sufficient.

Hence, (B).

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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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mmphf wrote:
Is |x-y|>|x|-|y| ?

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


Responding to a pm:
You can solve this question easily if you understand some basic properties of absolute values. They are discussed in detail here: http://www.veritasprep.com/blog/2014/02 ... -the-gmat/

One of the properties is

(II) For all real x and y, \(|x - y| \geq |x| - |y|\)
\(|x - y| = |x| - |y|\) when (1) x and y have the same sign and x has greater (or equal) absolute value than y (2) y is 0
\(|x - y| > |x| - |y|\) in all other cases

Question: Is |x-y|>|x|-|y|?

We need to establish whether the "equal to" sign can hold or not.

(1) y < x
Doesn't tell us whether they have the same sign or opposite. So we don't know whether the equal to sign will hold or greater than. Not sufficient.

(2) xy < 0
Tells us that one of x and y is positive and the other is negative (they do not have same sign). Also tells us that neither x nor y is 0. Hence, the "equal to" sign cannot hold. Sufficient to answer 'Yes'

Answer (B)
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New post 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| ?

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New post 17 Dec 2014, 04:50
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

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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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New post 17 Dec 2014, 07:08
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.
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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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New post 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

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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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New post 12 Jan 2015, 19:59
harsh01 wrote:
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


Note that "Is a > b" is not the same question as "Is a^2 > b^2?"

Say, a = 5 and b = -5

Is a > b? Yes
Is a^2 > b^2? No
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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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New post 30 May 2015, 16:30
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


You're forgetting that in statement 2, given that xy<0, either x or y need to be negative.

Scenario 1: x is negative
|-1/2 - 1/3| = |-5/6| = 5/6 ....other side of the equation....|-1/2|-|1/3| = 1/6
5/6 > 1/6

Scenario 2: y is negative
|1/2-(-1/3)| = 5/6....other side of the equation....|1/2|-|-1/3| = 1/2-1/3 = 1/6
5/6 > 1/6

In both scenarios |x-y| is greater than |x|-|y|. This would also be true if you were to switch the values so x = 1/3 and y = 1/2. So statement 2 is sufficient

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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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New post 27 Jul 2015, 15:25
For the algebraically inclined, here is how i solved it:

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

Simplifying the statement-

squaring both sides:
x^2 + y^2 - 2xy > x^2 + y^2 - 2|x||y|

Cancelling common terms:
-2xy > - 2|x||y|

Dividing by -2 on both sides:
xy < |x||y| ........ which implies xy < 0
So, B stands.

From the above derivation, it is clear that statement A -> y < x, as an independently does not suffice.

Hence, B is our answer.



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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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New post 11 Oct 2015, 07:17
mmphf wrote:
Is |x-y|>|x|-|y| ?

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



A great way to rember the rules Bunuel mentions for me is to remember what happens when you have a difference between a single absolute value term |X-Y| and a difference between two seperate absolute values |x| - |y|. So in order for |x-y| to be greater than |x| - |y|, xy need to have different signs.

Why is this? Because if xy<0, xy add to each other in every case within this term |x - y| while within |x| - |y| the whole value of the RHS shrinks.
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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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New post 14 Oct 2015, 18:47
Bunuel wrote:
someone79 wrote:
This is a very simple question. !x-y!>|x|-|y| can only happen if both the numbers are of different signs.

If xy<0 then these numbers are of opposite signs. Hope this clears.


X=2 y=3 then |x-y|=|x|-|y|
if x=-2 and y = 3 then |x-y|>|x|-|y|


Red part is not correct \(|x-y|>{|x|-|y|}\) also holds true when \(x\) and \(y\) have the same sign and the magnitude of \(y\) is more than that of \(x\) (so for \(|y|>|x|\)). Example:
\(x=2\) and \(y=3\) --> \(|x-y|=1>-1={|x|-|y|}\);
\(x=-2\) and \(y=-3\) --> \(|x-y|=1>-1={|x|-|y|}\).

Actually the only case when \(|x-y|>{|x|-|y|}\) does not hold true is when \(xy>{0}\) (so when \(x\) and \(y\) have the same sign) and \(|x|>|y|\) (simultaneously). In this case \(|x-y|={|x|-|y|}\) (as shown in my previous post). Example:
\(x=3\) and \(y=2\) --> \(|x-y|=1={|x|-|y|}\);
\(x=-3\) and \(y=-2\) --> \(|x-y|=1={|x|-|y|}\).

Hope it's clear.




Bunuel

This post has left me confused. Earlier for these questions I used to plug in values and got to know if the inequality is valid or not.

As per the algebraic method, this inequality stands when

1. If x and y have different signs (got this by the plug in method)
2. If x and y have same signs but y>x (above post)

So now, we need to look out for a scenario when x and y are positive and y>x. So had the second statement been xy>0 then the answer would have been C?

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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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longfellow wrote:
Bunuel wrote:
someone79 wrote:
This is a very simple question. !x-y!>|x|-|y| can only happen if both the numbers are of different signs.

If xy<0 then these numbers are of opposite signs. Hope this clears.


X=2 y=3 then |x-y|=|x|-|y|
if x=-2 and y = 3 then |x-y|>|x|-|y|


Red part is not correct \(|x-y|>{|x|-|y|}\) also holds true when \(x\) and \(y\) have the same sign and the magnitude of \(y\) is more than that of \(x\) (so for \(|y|>|x|\)). Example:
\(x=2\) and \(y=3\) --> \(|x-y|=1>-1={|x|-|y|}\);
\(x=-2\) and \(y=-3\) --> \(|x-y|=1>-1={|x|-|y|}\).

Actually the only case when \(|x-y|>{|x|-|y|}\) does not hold true is when \(xy>{0}\) (so when \(x\) and \(y\) have the same sign) and \(|x|>|y|\) (simultaneously). In this case \(|x-y|={|x|-|y|}\) (as shown in my previous post). Example:
\(x=3\) and \(y=2\) --> \(|x-y|=1={|x|-|y|}\);
\(x=-3\) and \(y=-2\) --> \(|x-y|=1={|x|-|y|}\).

Hope it's clear.




Bunuel

This post has left me confused. Earlier for these questions I used to plug in values and got to know if the inequality is valid or not.

As per the algebraic method, this inequality stands when

1. If x and y have different signs (got this by the plug in method)
2. If x and y have same signs but y>x (above post)

So now, we need to look out for a scenario when x and y are positive and y>x. So had the second statement been xy>0 then the answer would have been C?


Yes. If the 2nd statement had mentiond that xy>0 ---> this would mean that both x and y are of the same sign but without knowing whether y>x , this statement would not have been sufficient.

With statement 1, y<x and statement 2 (new) xy>0, you would have been able to answer the question unambiguously, leading to C.

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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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New post 15 Oct 2015, 17:58
tricky one, as one combination of X and Y according to first statement gives makes the equation equal to whereas in question it is asked greater than, that the point which differentiates answer B from D
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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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New post 19 Oct 2015, 22:55
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

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

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

If we modify the original condition,
if |x|>=|y|,
|x-y|>|x|-|y|>=0, so we can square both sides, which gives us (|x-y|)^2>(|x|-|y|)^2, and (x-y)^2>(|x|-|y|)^2
This becomes x^2+y^2-2xy>x^2+y^2-2|xy|, and if we simplify the inequality, -2xy>-2|xy|, and we ultimately want to know whether xy<0. This makes condition 2 sufficient.
if |x|<|y|, then |x|-|y|<0,
so |x-y|>|x|-|y| always work. Hence we do not need to deal with this, so the answer becomes (B).

Once we modify the original condition and the question according to the variable approach method 1, we can solve approximately 30% of DS questions.
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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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New post 28 Oct 2015, 10:14
Alternate method:

|X-Y| > |X| - |Y|?

|x-y| => distance of y from X
|x| => |x-0| => distance of x from zero
|y| => |y-0| => distance of y from zero

Statement 1)

y< x

------0------y----x
or
---y---0-------x----
or
----y------x-----0

distance still remains unknown ( not sufficient)

Statement 2)

----y---0----x---
or
--x-----0----y


Distance between x and y will always be greater than (distance between x and 0) - (distance between y and 0) = > because of additive effect in the former and diminishing effect of the latter

Choice: B
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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0 [#permalink]

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New post 23 Jan 2016, 09:12
mmphf wrote:
Is |x-y|>|x|-|y| ?

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


Let's just plug some numbers, I hate memorizing some complex formulas, in cases in which it's not needed.
(1) x=10, y=3 --> 7 > 7 NO, now, let's make y negative:
x=10, y=-3 --> 13 > 7 YES, two answers hence not sufficient

(2) two cases are possible: a) x- y+ b) x+ y-, --> there is no need to test case B, as we have already tested it above (x=10, y=-3 --> 13 > 7 YES)
So, let's test case A: x=-5, y=5 --> 10 > 0 YES, we are done. Sufficient.

Answer B
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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0   [#permalink] 23 Jan 2016, 09:12

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