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

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Re: Is |x - y|>|x - z|? (1) |y|>|z| (2) x < 0 [#permalink]  24 Jun 2012, 11:39
1
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kuttingchai wrote:
Voted for E

Solved it following way

(A) |Y| > |Z|

Case 1 :

---[-X =-4]-------[-Y = -3]------[-Z = -2]------[0]-------[Z = 2]-------[Y=3]-------[X=4]------

Case 2 :

---[-Y = -3]------[-Z = -2]------[-X =-1]-------[0]-------[X=1]-------[Z = 2]-------[Y=3]------

when x>0

|X-Y| = |1-3| = 2 and |X-Z| = |1-2| = 1
therefore |X-Y| > |X-Z|

|X-Y| = |4-3| = 1 and |X-Z| = |4-2| = 2
therefore |X-Y| < |X-Z|

when x<0

|X-Y| = |-1+3| = 2 and |X-Z| = |-1+2| = 1
therefore |X-Y| > |X-Z|

|X-Y| = |-4+3| = 1 and |X-Z| = |-4+2| = 2
therefore |X-Y| < |X-Z|

not sufficient

(B) X<0 dont know about Y and Z, therefore nit sufficient

(C) X < 0 and |Y| > |Z|

when x<0

|X-Y| = |-1+3| = 2 and |X-Z| = |-1+2| = 1
therefore |X-Y| > |X-Z|

|X-Y| = |-4+3| = 1 and |X-Z| = |-4+2| = 2
therefore |X-Y| < |X-Z|
[not sufficent]

therefore E

First, remember what do we use the absolute value for ? We use it to express the distance between two real numbers on the number line.
So, |x - y| means the distance between x and y, and |x| = |x - 0| means the distance between x and 0. Also, |x + 3| = |x - (-3)| expresses the distance between x and -3. Obviously, |x - y| = |y - x| (it is the same distance).

So, the question can be rephrased as "is the distance between x and y greater that the distance between x and z ?" or "is x closer to z than to y?"

(1) |y| > |z| means that the distance from y to 0 is greater than the distance from z to 0. This in fact is not important in this case. But certainly y and z are distinct. Regardless of wheather y > z or y < z (both cases are possible, try to draw the number line and visualize the points), we can place x closer to either y or z. So, (1) is not sufficient.

(2) Is evidently not sufficient. Take a point x on the number line at the left of 0, then you can place y and z as you please, each one can be closer or farther from x. Also, now you can consider y = z, in which case the two distances are equal.

Evidently, considering (1) and (2) together won't help either. For example, put x at the left of 0, y and z on either side of x such that y < x and z > x, both negative. You can play with each distance and put either y or z closer to x.

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Re: Is |x - y|>|x - z|? (1) |y|>|z| (2) x < 0 [#permalink]  27 Jun 2012, 06:52
Friends, Am new here. Just started to prepare. Please let me know if my thought process on this problem is right.

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

1) |Y| > |Z|.
--- > |Y| = +Y OR -Y
--- > |Z| = +Z OR -Z
Basis the above, |X-Y| = X-Y OR X+Y OR Y-X OR -Y-X
|X-Z| = X-Z OR X+Z OR Z-X OR -Z-X
Hence we are not sure which one out of the above matches to give a certain answer.

2) X<0
This by itself is not sufficient because we dont know the signs of Y and Z

1 + 2 , we know X is negative and if we consider the negative value of X in |X-Y| = X-Y OR X+Y OR Y-X OR -Y-X & |X-Z| = X-Z OR X+Z OR Z-X OR -Z-X , still we are unclear
which one will certainly answer the question. Hence E.

( pl direct me correctly , if Iam wrong)
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Re: Is |x - y|>|x - z|? (1) |y|>|z| (2) x < 0 [#permalink]  17 Jan 2013, 03:32
Hussain15 wrote:
Is |x - y|>|x - z|?

(1) |y|>|z|
(2) x < 0

My approach is to use the distance perspective...
|x-z| represents the distance bet. x and z
|x-y| represents the distance bet. x and y

1. |y| > |z|

<-----------------------0-------z------y------->
or
<--------------y--------0------z------------->
or
<--------------y-----z---0--------------------->
or
<--------------------z----0------------y------->

But we don't know where x lies... INSUFFICIENT

2. x < 0

<-------------x----------0------------------------>

But we dont know where y and z lie... INSUFFICIENT

Combine:

scenario1: <-----------------x------0-------z------y-------> |x-z| < |x-y|
scenario2: <---------y-------x------0-------z--------------> |x-z| > |x-y|

and many more scenarios... BUT two shows that info is NOT SUFFICIENT

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Re: Modules problem [#permalink]  28 Mar 2013, 01:20
1
KUDOS
Can anyone solve the following problem in a completely algebraic way? I mean, without plugging any numbers.
Is |x-z| > |x-y| ?

1. |z| > |y|
2. 0 > x

$$|x-z| > |x-y| = (x-z)^2>(x-y)^2 = x^2+z^2-2xz>x^2+y^2-2xy$$
$$2xy-2xz>y^2-z^2$$
$$2x(y-z)>(y+z)(y-z)$$

1. |z| > |y|
$$z^2>y^2$$
$$z^2-y^2>0=y^2-z^2<0=(y-z)(y+z)<0$$
So one between (y-z) (y+z) is negative, the other is positive, but we cannot tell which one is +ve or -ve.
$$2x(y-z)>(y+z)(y-z)$$
the second part is -ve ((y-z)(y+z)<0) we cannot say anything about 2x(y-z)
Not sufficient

2. 0 > x
x<0 so the first term is -ve (2x) but we cannot say anything about the other part (...(y-z)>(y+z)(y-z))

(1)+(2)
Still not sufficient, let me explain. Here are the combinations ( remeber that one between (y-z) (y+z) is negative and the other positive)
$$2x(y-z)>(y+z)(y-z)$$
Case one (y-z)-ve: 2x<0 (y-z)<0 (y+z)>0
-veNumber*-veNumber>+veNumber*-veNumber
+ve>-ve always true.
Case two (y+z)-ve: 2x<0 (y-z)>0 (y+z)<0
-veNumber*+veNumber>-veNumber*+veNumber
-veNumber>-veNumber we cannot say if this is true, since we have no numerical value

E

Do I deserve a Kudos for this?
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Re: Is |x - y|>|x - z|? (1) |y|>|z| (2) x < 0 [#permalink]  10 May 2013, 16:36
Hussain15 wrote:
Is |x - y|>|x - z|?

(1) |y|>|z|
(2) x < 0

Absolute Value of a number describe the distance of a number from 0 in a number line irrespective of a positive or negative number. Similarly |x-y| gives the distance between two numbers irrespective of their sign. Pick some numbers and you can reach this conclusion.

1)|y| > |z| -> just tells you y is further from z on the number line. It doesn't tell you where x is -> INSUFFICIENT.

2) x < 0. Again doesn't help because we are talking about distance between numbers. We don't any relation between x and y, hence INSUFFICIENT.

Combining 1 and 2 also doesn't give much information. We just know y is farther from z and x is negative. This can be interpreted in multiple ways.

Hence E.
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Re: Is |x - y|>|x - z|? (1) |y|>|z| (2) x < 0 [#permalink]  13 May 2013, 03:36
square both sides , question becomes is 2x(y+z)<0 , this is possible in 2 cases

a) x-ve and y+z +ve
b) x +ve and y+z -ve

from 1

y^2 - x2 >0 , i.e. (y-x)(y+x) > 0 ... no idea about x on its own , (y+x) we can never tell ... insuff

from 2

x is -ve .... no idea about ( y+x) ...... insuff

both

x is -ve so assessing whether (y+z) is +ve or not .... we cant tell.........E
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Re: Is |x - y|>|x - z|? (1) |y|>|z| (2) x < 0 [#permalink]  13 May 2013, 23:25
+1 Bunuel.....Nice explanation..I went to the extent of taking examples for each case and solving it..This graphical approach is awesome!!!!
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Re: Is |x - y|>|x - z|? (1) |y|>|z| (2) x < 0 [#permalink]  30 Jun 2013, 10:26
Is |x - y|>|x - z|?

first, let's square both sides:

Is (x-y)^2 > (x-z)^2
Is (x-y)*(x-y) > (x-z)*(x-z)
Is x^2 -2xy + y^2 > x^2 - 2xz + z^2
Is 2xz - 2xy > z^2 - y^2?
Is 2(xz-xy) > z^2 - y^2?
Is 2x(z-y) > (z+y)*(z-y)?

1. |z| > |y|
We can square both sides to factor out in an attempt to understand more about x and y [as we see in the stem - (z+y)*(z-y)]

z^2 - y^2 > 0
(z + y)*(z - y) > 0

So, (z+y)*(z-y) may be both positive or both negative for the product to be greater than zero. However, this tells us nothing about the left hand side.
INSUFFICIENT

2. 0 > x

This tells us the the left hand side of 2x(z-y) > (z+y)*(z-y) is positive, however, it tells us nothing about the right hand side.
INSUFFICIENT

1+2) We know that the RHS of 2x(z-y) > (z+y)*(z-y) is positive and we also know that 2x is positive as well. However, there is one problem. We have established that (z + y)*(z - y) is positive which means BOTH (z + y)*(z - y) are positive or BOTH (z + y)*(z - y) are negative. We don't know if they are positive or negative though, which means that 2x(z-y) could be positive or negative. This problem could also be tested by picking numbers.
INSUFFICIENT

(E)
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Re: Is |x - y|>|x - z|? (1) |y|>|z| (2) x < 0 [#permalink]  23 Jul 2013, 12:55
Is |x - y|>|x - z|?

(1) |y|>|z|

I. |4|>|3|
II. |4|>|-3|
III. |-4|>|3|
IV. |-4|>|-3|

This tells us nothing about x.
INSUFFICIENT

(2) x < 0
This tells us nothing about y and z.
INSUFFICIENT

1+2) |y|>|z| and x < 0
If the absolute value of y is greater than the absolute value of z and we are told that x < 0 (i.e. x is negative)

|x - y|>|x - z|
|-2 - (-4)| > |-2 - 3|
|2| > |-5|
2 > 5 Invalid

|x - y|>|x - z|
|-2 - 4|>|-2-3|
|-6|>|-5|
6>5 Valid

Even knowing that x is negative, we can't sufficently narrow down the signs of x and y to know if |x - y|>|x - z| holds true or not.
INSUFFICIENT

(E)
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Re: Is |x - y|>|x - z|? (1) |y|>|z| (2) x < 0 [#permalink]  21 Aug 2014, 11:09
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Re: Is |x - y|>|x - z|? (1) |y|>|z| (2) x < 0   [#permalink] 21 Aug 2014, 11:09

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