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Source: — Data Sufficiency |
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by clock60 » Sun May 15, 2011 12:46 pm
hi
If \frac{x}{|x|}<x which of the following must be true about x?
is it possible to write underlined part in some other way.i guess no one knows exactly what is written here
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by Ian Stewart » Mon May 16, 2011 1:54 pm
diebeatsthegmat wrote:If \frac{x}{|x|}<x which of the following must be true about x?

(A) x>1

(B) x>-1

(C) |x|<1

(D) |x|=1

(E) |x|^2>1

a is my answer choice. whats yours>?
The notation in the question "\frac{x}{|x|}<x" is used in a mathematical typesetting environment called TeX - it's not something you'll ever see on the GMAT. The question should begin:

If x/|x| < x, which of the following...

Now x/|x| is either equal to 1 (if x is positive) or to -1 (if x is negative). So we can rephrase the inequality:

* if x > 0, then 1 < x
* if x < 0, then -1 < x

So there are two ranges of possible values for x; either x > 1, or -1 < x < 0.

From here you can see that the only answer which *must* be true is B, since if x is in either of the ranges above, x is certainly greater than -1. Most of the other answers (besides D) *could* be true, but are not always true.
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by sourabh33 » Mon May 16, 2011 3:48 pm
Hi Ian

I have a small doubt regarding statements A & B.

I can't think of any value of x>1 that will not be true for the equation x/|x| < X
Versus For option B (x>-1), when x = 0, then the equation x/|x| < X may not be true

Please help.
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by Ian Stewart » Mon May 16, 2011 5:27 pm
sourabh33 wrote:Hi Ian

I have a small doubt regarding statements A & B.

I can't think of any value of x>1 that will not be true for the equation x/|x| < X
Versus For option B (x>-1), when x = 0, then the equation x/|x| < X may not be true

Please help.
I think you must be looking at the question backwards. We *know* that x/|x| is less than x. We want to know which answer choice *must* be true. It doesn't need to be true that x > 1, because x could be equal to -0.5, so A cannot be the right answer.

I think you're trying to answer the opposite question. That is, you're assuming that answer A is true, and you're trying to prove that x/|x| is less than x. It certainly is if x > 1, but that doesn't make A correct here, because that's not what the question is asking you to do.
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by sourabh33 » Mon May 16, 2011 6:04 pm
Thank you so much for the clarification.
Ian Stewart wrote: * if x > 0, then 1 < x
* if x < 0, then -1 < x

So there are two ranges of possible values for x; either x > 1, or -1 < x < 0.
But, as you said in your post that two ranges possible for x are x>1 and -1<x<0, than x cannot be 0 because either x has to be greater than 1 or x has to be in between -1 and 0.

Now in statement x>-1

In Range -1<x<0 the given equation is true
In Range x=0 the given equation is not defined
In Range 0<x<1 the given equation x/|x| is not true (it is actually greater than x (0.5/0.5 > 0.5))
In Range x=1 the equation is not true
In Range 1<x the equation is true
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