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If x is not equal to 0, is |x| less than 1? (1) x/|x|< x (2) |x| > x

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Re: If x is not equal to 0, is |x| less than 1? (1) x/|x|< x (2) |x| > x  [#permalink]

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New post 27 Dec 2017, 02:15
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Similar topic: https://gmatclub.com/forum/if-x-is-diff ... 26715.html
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Re: If x is not equal to 0, is |x| less than 1? (1) x/|x|< x (2) |x| > x  [#permalink]

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New post 29 Mar 2018, 05:33
I did it this way.

Is |x| < 1?

Statement 1

x/|x| < x

For, x>1
Divide by x on both sides, without changing the sign.
1/|x| < 1
|x| > 1

For x<1
Divide by x on both sides, and change the sign.
1/|x| > 1
|x| < 1

Insufficient, as we have both |x| > 1 (x is positive) and |x| < 1 (x is negative)

Statement 2

|x| > x
This is possible only when x < 0. (x is negative)

This is still insufficient to prove the given question.

Taking them both,

From (2), we get (x is negative) and from (1), the corresponding answer is |x| < 1.

Hence, C.


Hussain15 wrote:
If x is not equal to 0, is |x| less than 1?

(1) x/|x|< x

(2) |x| > x

PLEASE READ THE WHOLE DISCUSSION
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Re: If x is not equal to 0, is |x| less than 1? (1) x/|x|< x (2) |x| > x  [#permalink]

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New post 02 May 2018, 08:24
cruiseav wrote:
Bunuel wrote:
If x is not equal to 0, is |x| less than 1?

Is \(|x|<1\)?
Is \(-1<x<1\)? (\(x\neq{0}\))
So, the question asks whether x is in the range shown below:
Image


(1) \(\frac{x}{|x|}< x\)

Two cases:
A. \(x<0\) --> \(\frac{x}{-x}<x\) --> \(-1<x\). But remember that \(x<0\), so \(-1<x<0\)

B. \(x>0\) --> \(\frac{x}{x}<x\) --> \(1<x\).

Two ranges \(-1<x<0\) or \(x>1\). Which says that \(x\) either in the first range or in the second. Not sufficient to answer whether \(-1<x<1\). (For instance \(x\) can be \(-0.5\) or \(3\))

Second approach: look at the fraction \(\frac{x}{|x|}\) it can take only two values:
1 for \(x>0\) --> so we would have: \(1<x\);
Or -1 for \(x<0\) --> so we would have: \(-1<x\) and as we considering the range for which \(x<0\) then completer range would be: \(-1<x<0\).

The same two ranges: \(-1<x<0\) or \(x>1\):
Image


(2) \(|x| > x\). Well this basically tells that \(x\) is negative, as if x were positive or zero then \(|x|\) would be equal to \(x\). Only one range: \(x<0\), but still insufficient to say whether \(-1<x<1\). (For instance \(x\) can be \(-0.5\) or \(-10\))

Or consider two cases again:
\(x<0\)--> \(-x>x\)--> \(x<0\).
\(x>0\) --> \(x>x\): never correct.
Image


(1)+(2) Intersection of the ranges from (1) and (2) is the range \(-1<x<0\) (\(x<0\) (from 2) and \(-1<x<0\) or \(x>1\) (from 1), hence \(-1<x<0\)):
Image

Every \(x\) from this range is definitely in the range \(-1<x<1\). So, we have a definite YES answer to the question. Sufficient.


Answer: C.


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Attachment:
MSP8506173e0f30207c4a2700005eg6g59c48d51i5b.gif

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MSP111401047g43cgaehag920000640367fgf148e668.gif


Hi Bunuel

Surely i am missing something here..my basic doubt is in scenario when we take X<0

If we assume, x<0, then shouldn't the statement x/|x|<x = -x/-x <-x (my thinking here is if we considering x<0, then x should be negative throughout )
So, the equation becomes 1<-x = -1>x, we can keep this option as we have initially assumed X<0
Then if we take any values of x less then -1, |x| will always be greater than 1

if we assume X>0, then x/|x|<x = x/x<x.....which is 1<x, we can keep this option as initially we have assumed X>0
Then we take any values of x greater than 1 then |x| will always be greater than 1.

So statement 1 in either case is sufficient..

Thanks in advance for your help..


Negative x does not mean that you should replace x with -x. x just represents a negative number. You should replace |x| with -x, though because if x < 0, then |x| = -x.
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Re: If x is not equal to 0, is |x| less than 1? (1) x/|x|< x (2) |x| > x  [#permalink]

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New post 03 May 2018, 07:37
Bunuel wrote:
cruiseav wrote:
Bunuel wrote:
If x is not equal to 0, is |x| less than 1?

Is \(|x|<1\)?
Is \(-1<x<1\)? (\(x\neq{0}\))
So, the question asks whether x is in the range shown below:
Image


(1) \(\frac{x}{|x|}< x\)

Two cases:
A. \(x<0\) --> \(\frac{x}{-x}<x\) --> \(-1<x\). But remember that \(x<0\), so \(-1<x<0\)

B. \(x>0\) --> \(\frac{x}{x}<x\) --> \(1<x\).

Two ranges \(-1<x<0\) or \(x>1\). Which says that \(x\) either in the first range or in the second. Not sufficient to answer whether \(-1<x<1\). (For instance \(x\) can be \(-0.5\) or \(3\))

Second approach: look at the fraction \(\frac{x}{|x|}\) it can take only two values:
1 for \(x>0\) --> so we would have: \(1<x\);
Or -1 for \(x<0\) --> so we would have: \(-1<x\) and as we considering the range for which \(x<0\) then completer range would be: \(-1<x<0\).

The same two ranges: \(-1<x<0\) or \(x>1\):
Image


(2) \(|x| > x\). Well this basically tells that \(x\) is negative, as if x were positive or zero then \(|x|\) would be equal to \(x\). Only one range: \(x<0\), but still insufficient to say whether \(-1<x<1\). (For instance \(x\) can be \(-0.5\) or \(-10\))

Or consider two cases again:
\(x<0\)--> \(-x>x\)--> \(x<0\).
\(x>0\) --> \(x>x\): never correct.
Image


(1)+(2) Intersection of the ranges from (1) and (2) is the range \(-1<x<0\) (\(x<0\) (from 2) and \(-1<x<0\) or \(x>1\) (from 1), hence \(-1<x<0\)):
Image

Every \(x\) from this range is definitely in the range \(-1<x<1\). So, we have a definite YES answer to the question. Sufficient.


Answer: C.


Attachment:
MSP340910303aagbh2c14gf0000420i437if9d12bhg.gif

Attachment:
MSP2380151735322d2i2d790000280i6giebh17fdf3.gif

Attachment:
MSP8506173e0f30207c4a2700005eg6g59c48d51i5b.gif

Attachment:
MSP111401047g43cgaehag920000640367fgf148e668.gif


Hi Bunuel

Surely i am missing something here..my basic doubt is in scenario when we take X<0

If we assume, x<0, then shouldn't the statement x/|x|<x = -x/-x <-x (my thinking here is if we considering x<0, then x should be negative throughout )
So, the equation becomes 1<-x = -1>x, we can keep this option as we have initially assumed X<0
Then if we take any values of x less then -1, |x| will always be greater than 1

if we assume X>0, then x/|x|<x = x/x<x.....which is 1<x, we can keep this option as initially we have assumed X>0
Then we take any values of x greater than 1 then |x| will always be greater than 1.

So statement 1 in either case is sufficient..

Thanks in advance for your help..


Negative x does not mean that you should replace x with -x. x just represents a negative number. You should replace |x| with -x, though because if x < 0, then |x| = -x.


Hi Bunuel

Thanks for your reply. But i am still not able to get why we shouldn't consider negative x's throughout the equation.
If we assuming |x|= -x, we are assuming x<0, which means we are assuming the variable "X" as negative . So, we consider it negative only for Mode but keep it positive in other parts of the equation it seems inconsistent

Thanks again..
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Re: If x is not equal to 0, is |x| less than 1? (1) x/|x|< x (2) |x| > x  [#permalink]

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New post 03 May 2018, 08:53
cruiseav wrote:
Hi Bunuel

Thanks for your reply. But i am still not able to get why we shouldn't consider negative x's throughout the equation.
If we assuming |x|= -x, we are assuming x<0, which means we are assuming the variable "X" as negative . So, we consider it negative only for Mode but keep it positive in other parts of the equation it seems inconsistent

Thanks again..


Consider simple example: x = -1. So, we know that x is negative. Do you replace x there by -x and write -x = -1?
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Re: If x is not equal to 0, is |x| less than 1? (1) x/|x|< x (2) |x| > x  [#permalink]

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Re: If x is not equal to 0, is |x| less than 1? (1) x/|x|< x (2) |x| > x   [#permalink] 13 Mar 2020, 00:52

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