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Intern  Joined: 24 May 2014
Posts: 13
Location: Brazil
Is |x| < 1 ?  [#permalink]

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25 00:00

Difficulty:   95% (hard)

Question Stats: 40% (02:22) correct 60% (02:21) wrong based on 339 sessions

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Is |x| < 1 ?

(1) x/|x| < x

(2) x/|x| < 1

Originally posted by Reni on 31 Jul 2014, 09:55.
Last edited by Bunuel on 31 Jul 2014, 10:12, edited 1 time in total.
Edited the question.
Math Expert V
Joined: 02 Sep 2009
Posts: 58446
Re: Is |x| < 1 ?  [#permalink]

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7
5
Is |x| < 1 ?

Is |x| < 1 --> is -1 < x < 1.

(1) x/|x| < x.

Two cases:
A. $$x<0$$ --> $$\frac{x}{-x}<x$$ --> $$-1<x$$. Since we consider the case when $$x<0$$, then we'd have $$-1<x<0$$. Answer YES.

B. $$x>0$$ --> $$\frac{x}{x}<x$$ --> $$1<x$$. Answer NO.

Not sufficient.

(2) x/|x| < 1. This simply means that x is a negative number: if x is positive then x/|x| = 1, if x is negative x/|x| = -1 < 1. Not sufficient.

(1)+(2) Since from (2) we have that x is negative, then we have case A from (1), which gives an YES answer to the question. Sufficient.

Please tag the questions properly (this is NOT an algebra question) and copy the questions EXACTLY as they appear in the source.

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Manager  Joined: 31 Jul 2014
Posts: 126
GMAT 1: 630 Q48 V29 Re: Is |x| < 1 ?  [#permalink]

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Bunuel wrote:
Is |x| < 1 ?

Is |x| < 1 --> is -1 < x < 1.

(1) x/|x| < x.

Two cases:
A. $$x<0$$ --> $$\frac{x}{-x}<x$$ --> $$-1<x$$. Since we consider the case when $$x<0$$, then we'd have $$-1<x<0$$. Answer YES.

B. $$x>0$$ --> $$\frac{x}{x}<x$$ --> $$1<x$$. Answer NO.

Not sufficient.

(2) x/|x| < 1. This simply means that x is a negative number: if x is positive then x/|x| = 1, if x is negative x/|x| = -1 < 1. Not sufficient.

(1)+(2) Since from (2) we have that x is negative, then we have case A from (1), which gives an YES answer to the question. Sufficient.

Please tag the questions properly (this is NOT an algebra question) and copy the questions EXACTLY as they appear in the source.

Hi Bunuel
I fail to understand above solution.
We have to find whether -1< x<1
From 1 ) we know that x/|x| < x holds true for -1<x<0 and x>1 so we know that x is not in range 0<x<1
is this not sufficient to answer NO ....?
Again from 2) We know -infinity < x<0 then answer is NO
Sorry, could you please tell me what am I missing...how are you getting YES and NO for each?
CEO  S
Joined: 20 Mar 2014
Posts: 2597
Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44 GPA: 3.7
WE: Engineering (Aerospace and Defense)
Re: Is |x| < 1 ?  [#permalink]

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1
Bunuel wrote:
Is |x| < 1 ?

Is |x| < 1 --> is -1 < x < 1.

(1) x/|x| < x.

Two cases:
A. $$x<0$$ --> $$\frac{x}{-x}<x$$ --> $$-1<x$$. Since we consider the case when $$x<0$$, then we'd have $$-1<x<0$$. Answer YES.

B. $$x>0$$ --> $$\frac{x}{x}<x$$ --> $$1<x$$. Answer NO.

Not sufficient.

(2) x/|x| < 1. This simply means that x is a negative number: if x is positive then x/|x| = 1, if x is negative x/|x| = -1 < 1. Not sufficient.

(1)+(2) Since from (2) we have that x is negative, then we have case A from (1), which gives an YES answer to the question. Sufficient.

Please tag the questions properly (this is NOT an algebra question) and copy the questions EXACTLY as they appear in the source.

Hi Bunuel
I fail to understand above solution.
We have to find whether -1< x<1
From 1 ) we know that x/|x| < x holds true for -1<x<0 and x>1 so we know that x is not in range 0<x<1
is this not sufficient to answer NO ....?
Again from 2) We know -infinity < x<0 then answer is NO
Sorry, could you please tell me what am I missing...how are you getting YES and NO for each?

Let me try to explain.

From 1,

x/|x| < x ---> 2 cases:

--- when x $$\geq$$0 --> |x| = x ---> x/x < x --> x > 1 . Thus a "no" for is |x| < 1

--- when x < 0 --> |x| = -x ---> x/-x < x --> x > -1 . For x = 0.5, then a "yes" for |x|<1 but if x = 5, then a "no" for is |x|<1. Thus NOT sufficient.

From 2,

x/|x| < 1 , again 2 cases,

--- when x $$\geq$$0 --> |x| = x ---> x/x < 1 --> 1<1 . Thus x $$\geq$$0 is not a possible scenario.

--- when x <0 --> |x| = -x ---> x/-x < 1 --> 1>-1 . This is true for ALL x and thus x <0 satisfies this. For x = -0.5, you get a "yes" for is |x|<1 but for x = -4, you get a "no" for is |x| <1 . Thus NOT sufficient.

Alternately, if you want you can also test cases to come up with ranges for x. But this might be a bit more time consuming.

For your analysis of statement 1, the text in red is not correct. Refer to the explanation mentioned above.

For statement 2, you are correct that x<0 but you need to check whether -1<x<0 is satisfied or not. You get 2 different answers when you take x = -0.5 or x = -4.

Hope this helps.
Intern  Joined: 01 Jul 2015
Posts: 11
Is |x| < 1 ?  [#permalink]

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1
I have tried to use the line method to solve this problem.
1. x/!x!<xSince !x! is always positive, I can multiply it each side and get x<x.!x! and after substracting x-x.!x!<0
Now taking x as common factor, x(1-!x!)<0
x will have three factors:0,1,-1
Now refer to the attached figure to get the region where the given condition will be true.
Since the inequality is of "less than 0" type, we shold consider "negative portion"
we will get two possibilities which are -1<x<0 and x>1. Hence not suffecient.

2. x/!x!<1. Repeating the same logic above we get x-!x!<0. This will be true only if x<0. Still not suffeceint.

Combining both statements, we can eliminate the portion x>1, which we got in statement 1.
Leaving us only with -1<x<0, which tells us that !x! will always be a fraction less than 1. Hence suffeceint.
Attachments 1.JPG [ 3.48 KiB | Viewed 3586 times ]

Manager  B
Joined: 19 Aug 2016
Posts: 75
Re: Is |x| < 1 ?  [#permalink]

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Engr2012 wrote:
Bunuel wrote:
Is |x| < 1 ?

Is |x| < 1 --> is -1 < x < 1.

(1) x/|x| < x.

Two cases:
A. $$x<0$$ --> $$\frac{x}{-x}<x$$ --> $$-1<x$$. Since we consider the case when $$x<0$$, then we'd have $$-1<x<0$$. Answer YES.

B. $$x>0$$ --> $$\frac{x}{x}<x$$ --> $$1<x$$. Answer NO.

Not sufficient.

(2) x/|x| < 1. This simply means that x is a negative number: if x is positive then x/|x| = 1, if x is negative x/|x| = -1 < 1. Not sufficient.

(1)+(2) Since from (2) we have that x is negative, then we have case A from (1), which gives an YES answer to the question. Sufficient.

Please tag the questions properly (this is NOT an algebra question) and copy the questions EXACTLY as they appear in the source.

Hi Bunuel
I fail to understand above solution.
We have to find whether -1< x<1
From 1 ) we know that x/|x| < x holds true for -1<x<0 and x>1 so we know that x is not in range 0<x<1
is this not sufficient to answer NO ....?
Again from 2) We know -infinity < x<0 then answer is NO
Sorry, could you please tell me what am I missing...how are you getting YES and NO for each?

Let me try to explain.

From 1,

x/|x| < x ---> 2 cases:

--- when x $$\geq$$0 --> |x| = x ---> x/x < x --> x > 1 . Thus a "no" for is |x| < 1

--- when x < 0 --> |x| = -x ---> x/-x < x --> x > -1 . For x = 0.5, then a "yes" for |x|<1 but if x = 5, then a "no" for is |x|<1. Thus NOT sufficient.

From 2,

x/|x| < 1 , again 2 cases,

--- when x $$\geq$$0 --> |x| = x ---> x/x < 1 --> 1<1 . Thus x $$\geq$$0 is not a possible scenario.

--- when x <0 --> |x| = -x ---> x/-x < 1 --> 1>-1 . This is true for ALL x and thus x <0 satisfies this. For x = -0.5, you get a "yes" for is |x|<1 but for x = -4, you get a "no" for is |x| <1 . Thus NOT sufficient.

Alternately, if you want you can also test cases to come up with ranges for x. But this might be a bit more time consuming.

For your analysis of statement 1, the text in red is not correct. Refer to the explanation mentioned above.

For statement 2, you are correct that x<0 but you need to check whether -1<x<0 is satisfied or not. You get 2 different answers when you take x = -0.5 or x = -4.

Hope this helps.

Statement 2

Aren't we told that x lies between 1 and -1 ?

then why are we considering x=-4

Pls help

Senior Manager  P
Joined: 10 Apr 2018
Posts: 266
Location: United States (NC)
Re: Is |x| < 1 ?  [#permalink]

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Hi,

There is similar question out there
https://gmatclub.com/forum/if-x-is-not- ... 86140.html

Now here Bunuel has explained two approaches for solving the question
https://gmatclub.com/forum/if-x-is-not- ... 86140.html

Hope this helps

Probus
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Probus

~You Just Can't beat the person who never gives up~ Babe Ruth
SVP  P
Joined: 03 Jun 2019
Posts: 1724
Location: India
Re: Is |x| < 1 ?  [#permalink]

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Reni wrote:
Is |x| < 1 ?

(1) x/|x| < x

(2) x/|x| < 1

Is |x| < 1 ?

(1) x/|x| < x
$$\frac{x}{|x|}- x <0$$
$$\frac{x - x|x|}{|x|} < 0$$
Since |x| > 0
x - x|x| < 0
x (1- |x|) < 0
x (|x| -1) > 0
If x>0; |x| > 1; x>1
If x<0; |x| < 1; -1<x<0
Either x>1 or -1<x<0
NOT SUFFICIENT

(2) x/|x| < 1
x/|x| - 1 < 0
x - |x| < 0
If x>0; x-x<0;
If x<0; x + x < 0; x<0
|x| may or may not be <1
NOT SUFFICIENT

(1) + (2)
(1) x/|x| < x
If x>0; |x| > 1; x>1
If x<0; |x| < 1; -1<x<0
(2) x/|x| < 1
x<0
-1<x<0
|x|<1
SUFFICIENT

IMO C
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E-mail : kinshook.chaturvedi@gmail.com Re: Is |x| < 1 ?   [#permalink] 24 Sep 2019, 09:10
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