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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.

Therefore, answer, E. _________________

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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.

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------->

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

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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.

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

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

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