Inequality

[justdabz]

Is |x-z|>|x-y|?
1). |z|>|y|
2). 0>x

Please explain (without plugging )?

Source: Testmagic forum

[JonathanSchneider]

There are a number of ways to approach this, but I'm going to put one forth here that is new to me (I usually try to solve these more algebraically):

The question is really asking us: "is the distance between x and z greater than the distance between x and y?" If we take x, y, and z to be numbers on a number line, there are six possible arrangements (easily given by 3!):

x y z
x z y
y x z
y z x
z x y
z y x

Now, we can see that in the first and last cases x is clearly farther away from z than it is from y, making the answer to this question a distinct "yes." In the second and fourth cases, meanwhile, x is clearly farther away from y than it is from z, making the answer to this question a distinct "no." Note that this is the case regardless of where x, y, and z are situated on the number line. (Of course, this presupposes that x, y, and z all have different values. A real GMAT problem of this complexity might tell us this much as a starting point.) Now, as for cases 3 and 5, here x is situated in between y and z, so we do not know whether x is closer to one or the other.

From the above, the question becomes: do we know the arrangement of x, y, and z? (Or, more specifically, do we know that the arrangement is either: (a) case 1 or 6; or (b) case 2 or 4?)

Statement 2 tells us nothing about y or z, so it cannot be sufficient. Thus, B and D are out.

Statement 1 tells us only that z has a larger absolute value than y. However, this does not tell us the order of y and z, as either one could come first.

Together, we still do not have enough information. Although we know that x is negative, we do not know where to plot y and z.

[sanjaysingh82]

we need to know distance b/w x,z is greater or smaller than distance b/w xy

from stmt2 we only know thant x <o or --------------x-----0-------
from stm1 we know mod z > mod y but by how much we dont know.

hence E

[RonPurewal]

Is |x-z|>|x-y|?
1). |z|>|y|
2). 0>x

Please explain (without plugging )?

Source: Testmagic forum

jonathan's method will work, but a spatial understanding is much faster. the poster above me has the right idea, although his notes aren't very detailed; here is some more detail.

* the ABSOLUTE VALUE OF A DIFFERENCE is the DISTANCE between the two things in the difference.

so, |x - z| is just the distance between x and z on the number line, and |x - y| is just the distance between x and y on the number line.

therefore, the question is asking whether x is farther away from y than from z.

--

statement (1)
no information about the location of x at all; insufficient.

statement (2)
no information about the location of y or z; insufficient.

NOTE THAT YOU CAN GET DOWN TO C/E VERY QUICKLY. lots of hard problems are like this, actually: narrowing to 2 or 3 choices is easy, but going from there is much more difficult.

together:
|z| is |z - 0|, the distance between z and 0.
|y| is |y - 0|, the distance between y and 0.
so, statement (1) means that z is farther from 0 than y is.
this is one of 4 cases:
---z--y--0----
---z-----0--y-----
-------y--0-----z--
----------0--y--z--
we know only that x is negative; from these cases it's clear that x could be closer to y, closer to z, or equidistant from the two.
so, still way insufficient.

ans = (e)