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Manager  Joined: 05 Oct 2008
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Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0  [#permalink]

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

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

Attachment: 3_DS_Absolute_B.JPG [ 56.35 KiB | Viewed 86291 times ]
Math Expert V
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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

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.

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

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13
9

Here's how I did it.

|x-y| has a range of possible values , min being |x|-|y| and max being |x|+|y|

Statement 1 : x>y . Scenarioes :- x=+ve , y=-ve , |x-y|= |x|+|y| ;
x=+ve , y=+ve , |x-y|=|x|-|y| ;
x=-ve , y=-ve , |x-y|= |-(x-y)|=|x|-|y|

So we cannot definitely say that |x-y| is greater than |x|-|y| because the min value for |x-y| is also |x|-|y|. So, statement 1 is not sufficient.

Statement 2 : xy<0 . Scenarios :- x=+ve , y=-ve , |x-y|= |x|+|y|;
x=-ve , y=+ve , |x-y| = |-(|x|+|y|)|=|x|+|y|.

Now we know |x| + |y| is definitely greater than |x|-|y|. So statement (2) is sufficient.
##### General Discussion
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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0  [#permalink]

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Basically the question asks whether the distance between the two points x and y on the line is greater than the difference between the individual distances of x and y from 0.

$$|x-y|>|x|-|y|$$?

(1) $$y<x$$, 3 possible cases for $$|x-y|>|x|-|y|$$:

A. ---------------$$0$$---$$y$$---$$x$$---, $$0<y<x$$ --> in this case $$|x-y|>|x|-|y|$$ becomes: $$x-y>x-y$$ --> $$0>0$$. Which is wrong;
B. ---------$$y$$---$$0$$---------$$x$$---, $$y<0<x$$ --> in this case $$|x-y|>|x|-|y|$$ becomes: $$x-y>x+y$$ --> $$y<0$$. Which is right, as we consider the range $$y<0<x$$;
C. ---$$y$$---$$x$$---$$0$$--------------, $$y<x<0$$ --> in this case $$|x-y|>|x|-|y|$$ becomes: $$x-y>-x+y$$ --> $$x>y$$. Which is right, as we consider the range $$y<x<0$$.

(2) $$xy<0$$, means $$x$$ and $$y$$ have different signs, hence 2 cases for $$|x-y|>|x|-|y|$$:

A. ----$$y$$-----$$0$$-------$$x$$---, $$y<0<x$$ --> in this case $$|x-y|>|x|-|y|$$ becomes: $$x-y>x+y$$ --> $$y<0$$. Which is right, as we consider the range $$y<0<x$$;
B. ----$$x$$-----$$0$$-------$$y$$---, $$x<0<y$$ --> in this case $$|x-y|>|x|-|y|$$ becomes: $$-x+y>-x-y$$ --> $$y>0$$. Which is right, as we consider the range $$x<0<y$$.

In both cases inequality holds true. Sufficient.

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

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raghavs wrote:
is |x-y|>|x|-|y|

1>y<x
2>x*y<0

given expression, LHS = |x-y|; RHS = |x| - |y|

1) if y<x
Case I: x<0 => y<0 --> LHS = RHS
Case II: x>0, y>0 but <x --> LHS = RHS
Case III: x>0, y<0 --> LHS > RHS
Hence 1) alone is not sufficient

2) if x*y <0 => either x or y <0 and other has to be >0
Case I: x<0, y>0 --> LHS > RHS
Cae II: x>0, y<0 --> LHS > RHS
No other case.
Hence, 2) alone is sufficient

OA pls
Math Expert V
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Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0  [#permalink]

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

(1) $$y<x$$

Try two positive number $$x=3>y=1$$ --> is $$|3-1|>|3|-|1|$$? --> is $$2>2$$? Answer NO.

Try ANY other case but both positive: $$x=-5>y=-7$$ --> is $$|-5-(-7)|>|-5|-|-7|$$? --> is $$2>-2$$? Answer YES.

(2) $$xy<0$$, means $$x$$ and $$y$$ have different signs.

Now we can spot here that when $$x$$ and $$y$$ have different signs $$x-y$$ always "contribute" to each other so that its absolute value will increase: $$x=3$$, $$y=-1$$ --> $$|x-y|=|3+1|=|4|=4$$ or $$x=-3$$, $$y=1$$ --> $$|x-y|=|-3-1|=|4|=4$$.

But $$|x|-|y|$$ is difference (thus not "contributing") of two positive values (as neither equals to zero). $$x=3$$, $$y=-1$$ --> $$|x|-|y|=|3|-|-1|=2$$ or $$x=-3$$, $$y=1$$ --> $$|x|-|y|=|-3|-|1|=2$$.

So $$xy<0$$ means $$|x-y|>|x|-|y|$$ is always true.

Sufficient.

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

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Bunuel wrote:
Basically the question asks whether the distance between the two points x and y on the line is greater than the difference between the individual distances of x and y from 0.

$$|x-y|>|x|-|y|$$?

(1) $$y<x$$, 3 possible cases for $$|x-y|>|x|-|y|$$:

A. ---------------$$0$$---$$y$$---$$x$$---, $$0<y<x$$ --> in this case $$|x-y|>|x|-|y|$$ becomes: $$x-y>x-y$$ --> $$0>0$$. Which is wrong;
B. ---------$$y$$---$$0$$---------$$x$$---, $$y<0<x$$ --> in this case $$|x-y|>|x|-|y|$$ becomes: $$x-y>x+y$$ --> $$y<0$$. Which is right, as we consider the range $$y<0<x$$;
C. ---$$y$$---$$x$$---$$0$$--------------, $$y<x<0$$ --> in this case $$|x-y|>|x|-|y|$$ becomes: $$x-y>-x+y$$ --> $$x>y$$. Which is right, as we consider the range $$y<0<x$$.

(2) $$xy<0$$, means $$x$$ and $$y$$ have different signs, hence 2 cases for $$|x-y|>|x|-|y|$$:

A. ----$$y$$-----$$0$$-------$$x$$---, $$y<0<x$$ --> in this case $$|x-y|>|x|-|y|$$ becomes: $$x-y>x+y$$ --> $$y<0$$. Which is right, as we consider the range $$y<0<x$$;
B. ----$$x$$-----$$0$$-------$$y$$---, $$x<0<y$$ --> in this case $$|x-y|>|x|-|y|$$ becomes: $$-x+y>-x-y$$ --> $$y>0$$. Which is right, as we consider the range $$x<0<y$$.

In both cases inequality holds true. Sufficient.

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?

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

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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?

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

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4
Statement 1)
x>y.
therefore, x-y>0.
Plug & Play Method.
(x,y)- (-3,-6) .Satisfies.
(x,y)- (3,-6). Satisfies.
(x-y)- (3,6). Does not Satisfy. Equality exists.

Statement 2)
xy<0.
This means,
either, x>0 and y<0.
OR. x<0 and y>0.

Looking the values plugged in statement 1.
It satisfies the condition of statement two.
Hence,
B.
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GMAT 1: 670 Q42 V40 GMAT 2: 750 Q50 V42 Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0  [#permalink]

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Funny, but I remember form university that |a-b|>||a|-|b||>|a|-|b|, therefore the above inequality is valid for all numbers a,b can somebody verify the inequality?

http://math.ucsd.edu/~wgarner/math4c/de ... nequal.htm
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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0  [#permalink]

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Funny, but I remember form university that |a-b|>||a|-|b||>|a|-|b|, therefore the above inequality is valid for all numbers a,b can somebody verify the inequality?

http://math.ucsd.edu/~wgarner/math4c/de ... nequal.htm

If you notice, you have missed the 'equal to' sign.
Generalizing,$$|a-b|\geq|a|-|b|$$

In some cases, the equality will hold.
e.g. a = 3, b = 2
You get 1 = 1

In others, the inequality will hold.
e.g. a = -3, b = 4
7 > -1

In this question, you have to figure out whether the inequality will hold.
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Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0  [#permalink]

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

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

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(1) Y = -1, x = 0

Then, | 0 – (-1)| = 1
|0| - |-1| = -1

Y = 0, x = 1

Both |x-y| = |x| - |y| = 1

(2) Xy < 0, so one of them is < 0

So if we take the case x = -1, y = 1
Then |x – y| = |-2| = 2 and |x| - |y| = 1 – 1 = 0

Again, if x = 5 , y = -1
Then |x – y| = |6| = 6 and |x| - |y| = 5 – 1 = 4

So both 1 and 2 are insuff.

Combine them -> It is obvious that y < 0 and x > 0, so by adding a negative sign the magnitude increases and on the right side the magnitude will be less as the difference is between two positive numbers (i.e. the modulus values).

e.g. x = 2, y = -5

|x – y| = |7| and |x| - |y| = 2 – 5 = -3

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

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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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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

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|? (1) y < x (2) xy < 0  [#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

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|? (1) y < x (2) xy < 0  [#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

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|? (1) y < x (2) xy < 0  [#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

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|? (1) y < x (2) xy < 0  [#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

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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Originally posted by mau5 on 25 Mar 2013, 12:39.
Last edited by mau5 on 05 Apr 2013, 05:02, edited 1 time in total. Re: Is |x - y| > |x| - |y|? (1) y < x (2) xy < 0   [#permalink] 25 Mar 2013, 12:39

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