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Re: Is |x-y|>|x|-|y| [#permalink]

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New post 31 Mar 2012, 14:00
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I found talking through this one to be helpful.

Namely:

|x-y| represents the distance between x and y on the number line.

|x|-|y|, on the other hand, first takes the absolute value of both numbers - and thereby moving them both to the positive side of the number line - and THEN calculates the difference between x and y

Visually, it makes sense that if x and y are of different signs (for example, x=-5, y=5), then the difference between the two numbers on a number line is greater if measured before moving them both to the positive side of the number line.

At this point I logically deduced that it is impossible for |x-y| to be less than |x|-|y|. I also deduced at this point that if x and y have the same sign, it does not matter when the absolute value is taken because the difference between them will be the same either way.

After this thought process, the problem becomes much easier.
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Is |x-y|>|x|-|y| ? [#permalink]

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New post 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. :)

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 06 Mar 2013, 21:36
Answer: B

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.

So Answer is (B)
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Re: Is |x-y|>|x|-|y| [#permalink]

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New post 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]

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New post 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]

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New post 25 Mar 2013, 03:56
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]

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New post 25 Mar 2013, 04:29
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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 is B because if you use the data of statement 2, you can say, "Yes, |x-y| is greater than |x|-|y|"

(2) xy < 0
This means that one of x and y is positive and the other is negative. You cannot take x = 1/2 and y = 1/3.
It is not about fractions/integers. It is about positive/negative numbers (most mod questions are about positive/negative numbers)
When xy < 0, |x-y|>|x|-|y| always holds.
Only when x and y both are positive or both are negative and |x|>|y|, then |x-y|=|x|-|y|
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Re: Is |x-y|>|x|-|y| [#permalink]

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New post 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]

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New post 25 Mar 2013, 11:39
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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]

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New post 26 Mar 2013, 01:06
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]

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New post 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.

Answer: B.


(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]

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New post 05 Apr 2013, 01:55
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.

Answer: B.


(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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Re: Is |x-y|>|x|-|y| [#permalink]

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New post 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?)
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New post 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
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Re: Is |x-y|>|x|-|y| [#permalink]

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New post 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)
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Re: Is |x-y|>|x|-|y| [#permalink]

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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


Fractions and integers have no role to play here. Check Bunuel's post above.

Whenever xy < 0, i.e. x is negative but y is positive OR x is positive but y is negative, |x-y| is greater than |x|-|y|.

e.g. x = -1/2, y = 1/3
|x-y| = |-1/2-1/3| = 5/6
|x|-|y| = 1/2 - 1/3 = 1/6

So |x - y| > |x|-|y|

Do you see the logic here? If one of x and y is positive and the other is negative, in |x - y|, absolute values of x and y get added and the sum is positive. While in |x|-|y|, the absolute values are subtracted.
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Re: Is |x-y|>|x|-|y| [#permalink]

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New post 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.
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Re: Is |x - y| > |x| - |y|? [#permalink]

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New post 19 Dec 2013, 06:01
study wrote:
Is |x - y| > |x| - |y|?

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


I think the trick here is to remember the property
|x - y| >= |x| - |y|

Where = will only happen when both x and y have the same sign
Therefore, we need to discard that case

Statement 2 does it

Hope it helps
Cheers!
J :)
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Re: Is |x - y| > |x| - |y|? [#permalink]

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New post 01 Jan 2014, 20:55
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(1) is insuff. because if x and y equal to 10 and 2 respectively, then both sides of the inequality are the same weighted. But, if x, y are equal to 10 and -10 respectively, then left side is greater than the right side(from our perspective) of the inequality.

(2) states that either x or y is negative. We do not know which one is exactly negative. However, it is not important to find out this because in both cases the left side of the inequality is greater than the right side(from our perspective). Why?
It is simple. On the left side the numbers are added while on the right side the same numbers are subtracted from each other. Therefore, this statement is sufficient.

So, the correct answer is B.

Last edited by aja1991 on 02 Jan 2014, 02:36, edited 2 times in total.
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Re: gmat prep absolute values [#permalink]

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New post 02 Jan 2014, 20:39
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!
Re: gmat prep absolute values   [#permalink] 02 Jan 2014, 20:39

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