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If x/|x|<x which of the following must be true about x?

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Re: If x/|x|<x which of the following must be true about x? [#permalink]

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New post 21 Feb 2016, 14:46
Hi Karishma and other experts - I understand the solution but I'm still stuck on why reasoning is wrong. This is how I solved:

x/|x| < x --> x/|x| -x --> x*(1/|x|-1)<0, so:

1. x<0; (1/|x|-1)<0 -> 1/|x|<1 -> 1<|x| -> x<-1 THIS IS WHERE IT'S WRONG BUT WHY?

2. x>0; (1/|x|-1)>0 -> 1/|x|>1 -> 1>|x| -> 1>x THIS IS WHERE IT'S WRONG BUT WHY?

Thank you!
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Re: If x/|x|<x which of the following must be true about x? [#permalink]

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happyface101 wrote:
Hi Karishma and other experts - I understand the solution but I'm still stuck on why reasoning is wrong. This is how I solved:

x/|x| < x --> x/|x| -x --> x*(1/|x|-1)<0, so:

1. x<0; (1/|x|-1)<0 -> 1/|x|<1 -> 1<|x| -> x<-1 THIS IS WHERE IT'S WRONG BUT WHY?

2. x>0; (1/|x|-1)>0 -> 1/|x|>1 -> 1>|x| -> 1>x THIS IS WHERE IT'S WRONG BUT WHY?

Thank you!


The mistake you are making is in the portion with red text above. Your 'simplified' expression is x(1/|x|-1) and not just (1/|x|-1).

Thus your cases become:

Case 1: x<0 --> |x|=-x ---> x(1/|x|-1)<0 ---> x(-1/x-1)<0 ---->x(1/x+1)>0 --->x(1+x)/x > 0--->x+1>0 --->x>-1 and this with the fact that x<0 ---> range becomes -1<x<0.

Case 2: x>0 --> |x|=x ---> x(1/|x|-1)<0 ---> x(1/x-1)<0 ---->x(1/x-1)<0 --->x(1-x)/x < 0--->1-x<0 --->x>1 and this with the fact that x>0 ---> range becomes x>1.

Hope this helps.
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Re: If x/|x|<x which of the following must be true about x? [#permalink]

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New post 21 Feb 2016, 15:43
Engr2012 wrote:
happyface101 wrote:
Hi Karishma and other experts - I understand the solution but I'm still stuck on why reasoning is wrong. This is how I solved:

x/|x| < x --> x/|x| -x --> x*(1/|x|-1)<0, so:

1. x<0; (1/|x|-1)<0 -> 1/|x|<1 -> 1<|x| -> x<-1 THIS IS WHERE IT'S WRONG BUT WHY?

2. x>0; (1/|x|-1)>0 -> 1/|x|>1 -> 1>|x| -> 1>x THIS IS WHERE IT'S WRONG BUT WHY?

Thank you!


The mistake you are making is in the portion with red text above. Your 'simplified' expression is x(1/|x|-1) and not just (1/|x|-1).

Thus your cases become:

Case 1: x<0 --> |x|=-x ---> x(1/|x|-1)<0 ---> x(-1/x-1)<0 ---->x(1/x+1)>0 --->x(1+x)/x > 0--->x+1>0 --->x>-1 and this with the fact that x<0 ---> range becomes -1<x<0.

Case 2: x>0 --> |x|=x ---> x(1/|x|-1)<0 ---> x(1/x-1)<0 ---->x(1/x-1)<0 --->x(1-x)/x < 0--->1-x<0 --->x>1 and this with the fact that x>0 ---> range becomes x>1.

Hope this helps.


Thank you so much! +1
I think the knowledge gap is that I treated x*(1/|x|-1)<0 as if it's x*((1/|x|-1)=0
In an inequality problem I obviously can't solve by taking the two apart and do x<0, (1/|x|-1)<0 like I can for an equation. Thanks for helping me understand this!
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Re: If x/|x|<x which of the following must be true about x? [#permalink]

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happyface101 wrote:
Engr2012 wrote:
happyface101 wrote:
Hi Karishma and other experts - I understand the solution but I'm still stuck on why reasoning is wrong. This is how I solved:

x/|x| < x --> x/|x| -x --> x*(1/|x|-1)<0, so:

1. x<0; (1/|x|-1)<0 -> 1/|x|<1 -> 1<|x| -> x<-1 THIS IS WHERE IT'S WRONG BUT WHY?

2. x>0; (1/|x|-1)>0 -> 1/|x|>1 -> 1>|x| -> 1>x THIS IS WHERE IT'S WRONG BUT WHY?

Thank you!


The mistake you are making is in the portion with red text above. Your 'simplified' expression is x(1/|x|-1) and not just (1/|x|-1).

Thus your cases become:

Case 1: x<0 --> |x|=-x ---> x(1/|x|-1)<0 ---> x(-1/x-1)<0 ---->x(1/x+1)>0 --->x(1+x)/x > 0--->x+1>0 --->x>-1 and this with the fact that x<0 ---> range becomes -1<x<0.

Case 2: x>0 --> |x|=x ---> x(1/|x|-1)<0 ---> x(1/x-1)<0 ---->x(1/x-1)<0 --->x(1-x)/x < 0--->1-x<0 --->x>1 and this with the fact that x>0 ---> range becomes x>1.

Hope this helps.


Thank you so much! +1
I think the knowledge gap is that I treated x*(1/|x|-1)<0 as if it's x*((1/|x|-1)=0
In an inequality problem I obviously can't solve by taking the two apart and do x<0, (1/|x|-1)<0 like I can for an equation. Thanks for helping me understand this!


Additionally, you can split the factors as you did but you made an error there.

x*(1/|x|-1)<0
implies that x*(1/|x|-1) is negative.

So either x is negative and (1/|x|-1) is positive
or
x is positive and (1/|x|-1) is negative

Case 1: x is negative and (1/|x|-1) is positive
When x is negative, |x| = -x

(1/|x|-1) > 0
(1/(-x) - 1) > 0
1/x + 1 < 0
(1+x)/x < 0
-1 < x< 0

Case 2: x is positive and (1/|x|-1) is negative
When x is positive, |x| = x

(1/|x|-1) < 0
(1/x - 1) < 0
(1 - x)/x < 0
(x - 1)/x > 0

x > 1 or x < 0
But x is positive so x cannot be less than 0.
So x > 1.
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Re: If x/|x|<x which of the following must be true about x? [#permalink]

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New post 14 Mar 2016, 22:00
x/|x|<x which of the following must be true about x?

The range of values of x satisfying the above equation are shown in the image. -1<x<0 and x>1

(A) x>1 : From the figure, it is clear that for x>1 the given equation x/|x|<x always holds true. A may be the answer.

(B) x>−1: This range contains a set 0<x<1 which makes x/|x|<x not true. For Ex x=1/2 doesn't satisfy the equation.

(C) |x|<1: Contains x=1/2 which doesnot satisfy x/|x|<x .

(D) |x|=1: x=1 doesnot satisfy x/|x|<x .

(E) |x|2>1:This range contains set of values x<-1 which makes x/|x|<x not true. For Ex x=-4 doesn't satisfy the equation.

May plz guide me if not taken the question in right way.
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Re: If x/|x|<x which of the following must be true about x? [#permalink]

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New post 06 Jun 2016, 21:17
If the given answer B has to be correct. i.e., x > -1

then x = 1 should satisfy.

x/mod (x) < x
1/mod(1)<1
which is not correct.
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Re: If x/|x|<x which of the following must be true about x? [#permalink]

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New post 09 Jun 2016, 04:36
My previous concept got completed squashed when I solved this problem.

I always take the value which overlap. For example: x > 2 AND x < 7; overlapping area includes 3, 4, 5 and 6.

Similarly: for x>1 (when x is positive) OR −1<x<0 (when x is negative), I don't see any overlap. Am I missing the word OR here? Definitely not.

I am still clueless how come x>-1 is the solution for x>1 OR −1<x<0. If we take x>-1, then x can also be = 0 and in that case the inequality will be meaning less because we will have 0 in the denominator.

Bunnuel - Please help. Not sure what I am missing here.
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Re: If x/|x|<x which of the following must be true about x? [#permalink]

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New post 09 Jun 2016, 06:51
gauraku wrote:
My previous concept got completed squashed when I solved this problem.

I always take the value which overlap. For example: x > 2 AND x < 7; overlapping area includes 3, 4, 5 and 6.

Similarly: for x>1 (when x is positive) OR −1<x<0 (when x is negative), I don't see any overlap. Am I missing the word OR here? Definitely not.

I am still clueless how come x>-1 is the solution for x>1 OR −1<x<0. If we take x>-1, then x can also be = 0 and in that case the inequality will be meaning less because we will have 0 in the denominator.

Bunnuel - Please help. Not sure what I am missing here.


This question of yours will trip you on other MUST BE TRUE questions. A fool proof way to solve any MBT question is to realize that the correct answer will be true for ALL values while the incorrect options will fail at 1 or more than 1 possible values. I always recommend that for MBT questions, eliminate incorrect ones to arrive at the correct answer.

Coming back to this question, for your doubt x>-1 satisfies both the posisble ranges : -1<x<0 and x>1 and this is the reason why this option is the correct one. Other options are either partically covering these 2 ranges or none at all.

Let us try to use process of elimination (POE) to eliminate the other 4 incorrect options.

(A) \(x>1\) ---> what about x=-0.5, this value satisfies the given equation x/|x|<x but is not covered by this option. Eliminate.

(B) \(x>-1\)

(C) \(|x|<1\)---> this translates to -1<x<1, what about x=2, this value satisfies the given equation x/|x|<x but is not covered by this option. Eliminate.

(D) \(|x|=1\) ---> this translates to x= 1 or -1, what about x=-0.5, this value satisfies the given equation x/|x|<x but is not covered by this option. Eliminate.

(E) \(|x|^2>1\) ---> what about x=-0.5, this value satisfies the given equation x/|x|<x but is not covered by this option. Eliminate.

Thus, only option B is left and is thus the correct answer. I see your point that 0<x<1 isnt a correct range but you can surely say that if -1<x<0 OR x>1 then ALL values of x in these 2 ranges will be >-1. Thus x>-1 is the MBT condition for this question.

Hope this helps.
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Re: If x/|x|<x which of the following must be true about x? [#permalink]

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gauraku wrote:
My previous concept got completed squashed when I solved this problem.

I always take the value which overlap. For example: x > 2 AND x < 7; overlapping area includes 3, 4, 5 and 6.

Similarly: for x>1 (when x is positive) OR −1<x<0 (when x is negative), I don't see any overlap. Am I missing the word OR here? Definitely not.

I am still clueless how come x>-1 is the solution for x>1 OR −1<x<0. If we take x>-1, then x can also be = 0 and in that case the inequality will be meaning less because we will have 0 in the denominator.

Bunnuel - Please help. Not sure what I am missing here.


I have taken this issue in detail in this post: http://www.veritasprep.com/blog/2012/07 ... -and-sets/
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Re: If x/|x|<x which of the following must be true about x? [#permalink]

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New post 11 Jun 2016, 04:10
durgesh79 wrote:
i think OA is correct....

the question is If x / |x| < x. so we have to consider only those values of x for which this inequalty is true and what are those values
1. when x is between -1 and 0
2. when x is more than 1

lets call these conditions our universe.

Now the question is for all values of x (in our universe) which of the following is true
option B, x > -1, has both conditions 1 and 2

now you may say what about x = 1/2 .... that wasnt even part of our universe... so even if x = 1/2 is satisfying option B and not the question stem, we dont have to worry... becuase we are not supposed to take it as an example ...

for all values of x in our universe, option B is ALWAYS true...
Option A is not always true...


-1<x<0 and x>1 are the two areas where x can lie.
How can B be correct while we have answer choice like A(x>1) .
x>-1 tells that x can even be 0.5 but the range we found out does not say the same.
Plz clarify.
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Re: If x/|x|<x which of the following must be true about x? [#permalink]

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New post 12 Jun 2016, 20:53
sa18 wrote:
durgesh79 wrote:
i think OA is correct....

the question is If x / |x| < x. so we have to consider only those values of x for which this inequalty is true and what are those values
1. when x is between -1 and 0
2. when x is more than 1

lets call these conditions our universe.

Now the question is for all values of x (in our universe) which of the following is true
option B, x > -1, has both conditions 1 and 2

now you may say what about x = 1/2 .... that wasnt even part of our universe... so even if x = 1/2 is satisfying option B and not the question stem, we dont have to worry... becuase we are not supposed to take it as an example ...

for all values of x in our universe, option B is ALWAYS true...
Option A is not always true...


-1<x<0 and x>1 are the two areas where x can lie.
How can B be correct while we have answer choice like A(x>1) .
x>-1 tells that x can even be 0.5 but the range we found out does not say the same.
Plz clarify.


This same question has been asked many times in this thread and has been responded to time and again. You need to go through the thread. Right above your post, I have given the link where this concept is discussed in detail.

Also, focus on the question stem: which of the following must be true about x?
which means: which option must be true about ALL values that x can take?

Will ALL value of x be greater than 1? No. (A) incorrect.
Will ALL values of x be greater than -1? Yes. (B) correct.
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Re: If x/|x|<x which of the following must be true about x? [#permalink]

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New post 09 Mar 2017, 09:17
In option B, since it states x>-1, x could be 0.

Then the expression x/|x|<x will not stand true. Then how is option B the right answer?
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Re: If x/|x|<x which of the following must be true about x? [#permalink]

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New post 09 Mar 2017, 09:29
varunjoshi31 wrote:
In option B, since it states x>-1, x could be 0.

Then the expression x/|x|<x will not stand true. Then how is option B the right answer?


This issue has been addressed MANY times in previous 5 (!) pages. Please read the thread before posting a question. Thank you.
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Re: If x/|x|<x which of the following must be true about x? [#permalink]

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New post 15 Mar 2017, 15:41
nmohindru 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) \(|x|^2>1\)


We can safely say that x ≠ 0 since |x| is in the denominator. Since x ≠ 0:

1) x/|x| = 1 if x is positive, or

2) x/|x| = -1 if x is negative

Therefore, if x is positive, then 1 < x, i.e., x > 1, and if x is negative, then -1 < x, i.e., x > -1. So, we have x > 1 or x > -1. Since, if x is greater than 1, x is also greater than -1, choice B is the correct answer.

Answer: B
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Re: If x/|x|<x which of the following must be true about x? [#permalink]

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New post 16 Mar 2017, 07:28
Bunuel wrote:
nmohindru 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) \(|x|^2>1\)


This question was well explained by Durgesh and Ian Stewart, but since there are still some doubts, I'll try to add my 2 cents.

First of all let's solve this inequality step by step and see what is the solution for it, or in other words let's see in which ranges this inequality holds true.

Two cases for \(\frac{x}{|x|}<x\):

A. \(x<0\) --> \(|x|=-x\) --> \(\frac{x}{-x}<x\) --> \(-1<x\) --> \(-1<x<0\);

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

So given inequality holds true in the ranges: \(-1<x<0\) and \(x>1\). Which means that \(x\) can take values only from these ranges.

------{-1}xxxx{0}----{1}xxxxxx

Now, we are asked which of the following must be true about \(x\). Option A can not be ALWAYS true because \(x\) can be from the range \(-1<x<0\), eg \(-\frac{1}{2}\) and \(x=-\frac{1}{2}<1\).

Only option which is ALWAYS true is B. ANY \(x\) from the ranges \(-1<x<0\) and \(x>1\) will definitely be more the \(-1\), all "red", possible x-es are to the right of -1, which means that all possible x-es are more than -1.

Answer: B.


Hi Bunuel,

I have a doubt here. For 0<x<1, the condition does not hold. So why we are taking the range X > -1 instead of X > 1 when we know for X > -1, we also have 0<X<1 range.
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Re: If x/|x|<x which of the following must be true about x? [#permalink]

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New post 16 Mar 2017, 08:29
arunavamunshi1988 wrote:
Bunuel wrote:
nmohindru 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) \(|x|^2>1\)


This question was well explained by Durgesh and Ian Stewart, but since there are still some doubts, I'll try to add my 2 cents.

First of all let's solve this inequality step by step and see what is the solution for it, or in other words let's see in which ranges this inequality holds true.

Two cases for \(\frac{x}{|x|}<x\):

A. \(x<0\) --> \(|x|=-x\) --> \(\frac{x}{-x}<x\) --> \(-1<x\) --> \(-1<x<0\);

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

So given inequality holds true in the ranges: \(-1<x<0\) and \(x>1\). Which means that \(x\) can take values only from these ranges.

------{-1}xxxx{0}----{1}xxxxxx

Now, we are asked which of the following must be true about \(x\). Option A can not be ALWAYS true because \(x\) can be from the range \(-1<x<0\), eg \(-\frac{1}{2}\) and \(x=-\frac{1}{2}<1\).

Only option which is ALWAYS true is B. ANY \(x\) from the ranges \(-1<x<0\) and \(x>1\) will definitely be more the \(-1\), all "red", possible x-es are to the right of -1, which means that all possible x-es are more than -1.

Answer: B.


Hi Bunuel,

I have a doubt here. For 0<x<1, the condition does not hold. So why we are taking the range X > -1 instead of X > 1 when we know for X > -1, we also have 0<X<1 range.


This issue has been addressed MANY times in previous 5 (!) pages. Please read the thread before posting a question. Thank you.
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Re: If x/|x|<x which of the following must be true about x? [#permalink]

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New post 17 Mar 2017, 02:13
put test values as -2, -0.5, 0.5, 2
for -2, x/|x| >x
for -0.5, x/ |x| <x
for 0.5, x/|x| >x


thereore Option A i.e. x>1 holds true
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Re: If x/|x|<x which of the following must be true about x? [#permalink]

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Re: If x/|x|<x which of the following must be true about x?   [#permalink] 17 Mar 2017, 02:16

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If x/|x|<x which of the following must be true about x?

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