If x is not equal to 0, is |x| less than 1? : GMAT Data Sufficiency (DS)
Check GMAT Club Decision Tracker for the Latest School Decision Releases http://gmatclub.com/AppTrack

 It is currently 23 Jan 2017, 05:59

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# If x is not equal to 0, is |x| less than 1?

Author Message
TAGS:

### Hide Tags

Director
Joined: 01 Aug 2008
Posts: 768
Followers: 4

Kudos [?]: 626 [5] , given: 99

If x is not equal to 0, is |x| less than 1? [#permalink]

### Show Tags

06 Jan 2010, 12:31
5
KUDOS
8
This post was
BOOKMARKED
00:00

Difficulty:

95% (hard)

Question Stats:

43% (02:31) correct 57% (01:36) wrong based on 343 sessions

### HideShow timer Statistics

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

(1) x/|x| < x
(2) |x| > x
[Reveal] Spoiler: OA
Senior Manager
Joined: 22 Dec 2009
Posts: 362
Followers: 11

Kudos [?]: 378 [2] , given: 47

Re: |x| less than 1? [#permalink]

### Show Tags

06 Jan 2010, 13:00
2
KUDOS
ugimba wrote:
If x is not equal to 0, is |x| less than 1?

(1) $$x/|x| < x$$
(2) |x| > x

IMO A...

ST 1: can be written as $$x < x * |x|$$ since multiplying by |x| which is always positive would not change the sign.

This gives x as -1<x<0.... therefore sufficient...

ST2: |x| > x gives value of x as x<0... hence could be -2,-3.... Therefore not sufficient...

Cheers!
JT
_________________

Cheers!
JT...........
If u like my post..... payback in Kudos!!

|For CR refer Powerscore CR Bible|For SC refer Manhattan SC Guide|

~~Better Burn Out... Than Fade Away~~

Manager
Joined: 27 Aug 2009
Posts: 144
Followers: 2

Kudos [?]: 27 [0], given: 1

Re: |x| less than 1? [#permalink]

### Show Tags

06 Jan 2010, 14:27

X<X|x| ==> 1<|x| if X >0
==>1>|x| if X<0
so to conclude we need to know id X<0 or X>0

whats the OA??
Manager
Joined: 04 Feb 2007
Posts: 85
Followers: 2

Kudos [?]: 80 [2] , given: 16

Re: |x| less than 1? [#permalink]

### Show Tags

06 Jan 2010, 14:29
2
KUDOS
jeeteshsingh wrote:
ugimba wrote:
If x is not equal to 0, is |x| less than 1?

(1) $$x/|x| < x$$
(2) |x| > x

IMO A...

ST 1: can be written as $$x < x * |x|$$ since multiplying by |x| which is always positive would not change the sign.

This gives x as -1<x<0.... therefore sufficient...

ST2: |x| > x gives value of x as x<0... hence could be -2,-3.... Therefore not sufficient...

Cheers!
JT

1) this only holds true if x is a negative fraction or greater than 1. The question however is asking if |x| is less than 1. The abs val of a number less than -1 will be greater than 1, whereas the abs val of a negative fraction will be less than 1.

x/|x| < x

try -2, -2/|-2| = -1, -1 is greater than -2, so does not hold
try -1/2, (-1/2)/|(-1/2)| = -1, -1 is less than -1/2 so this holds.
try 1/2, (1/2)/|(1/2)| = 1, 1 is greater than 1/2, so this does not hold
try 2, 2/|2| = 1, 1 is less than 2, so this holds

so, -1 < x < 0 and x > 1

the question asks: is |x| < 1?

-1 < x < 0 --> |x| will be a positive fraction, i.e. less than 1
x > 1 --> |x| will be greater than 1

hence insufficient.

2) the abs val of a positive number equals that number so this only holds true for negative numbers (including fractions). The question is asking if |x| is less than 1.
Same as in (1),
|x| > x

x < 0

try a negative integer, -2,
| -2 | = 2, --> |x| will be greater 1

try a negative fraction, -(1/2),
| -(1/2) | = 1/2 --> |x| will be less than 1

hence insufficient.

putting both together,

x < 0 and -1< x < 0 and x > 1
this limits x to -1< x < 0, thus |x| will be a positive fraction and will always be less than 1.

_________________

If you like my post, a kudos is always appreciated

Senior Manager
Joined: 22 Dec 2009
Posts: 362
Followers: 11

Kudos [?]: 378 [2] , given: 47

Re: |x| less than 1? [#permalink]

### Show Tags

06 Jan 2010, 14:44
2
KUDOS
1
This post was
BOOKMARKED
firasath wrote:
jeeteshsingh wrote:
ugimba wrote:
If x is not equal to 0, is |x| less than 1?

(1) $$x/|x| < x$$
(2) |x| > x

IMO A...

ST 1: can be written as $$x < x * |x|$$ since multiplying by |x| which is always positive would not change the sign.

This gives x as -1<x<0.... therefore sufficient...

ST2: |x| > x gives value of x as x<0... hence could be -2,-3.... Therefore not sufficient...

Cheers!
JT

1) this only holds true if x is a negative fraction or greater than 1. The question however is asking if |x| is less than 1. The abs val of a number less than -1 will be greater than 1, whereas the abs val of a negative fraction will be less than 1.

x/|x| < x

try -2, -2/|-2| = -1, -1 is greater than -2, so does not hold
try -1/2, (-1/2)/|(-1/2)| = -1, -1 is less than -1/2 so this holds.
try 1/2, (1/2)/|(1/2)| = 1, 1 is greater than 1/2, so this does not hold
try 2, 2/|2| = 1, 1 is less than 2, so this holds

so, -1 < x < 0 and x > 1

the question asks: is |x| < 1?

-1 < x < 0 --> |x| will be a positive fraction, i.e. less than 1
x > 1 --> |x| will be greater than 1

hence insufficient.

2) the abs val of a positive number equals that number so this only holds true for negative numbers (including fractions). The question is asking if |x| is less than 1.
Same as in (1),
|x| > x

x < 0

try a negative integer, -2,
| -2 | = 2, --> |x| will be greater 1

try a negative fraction, -(1/2),
| -(1/2) | = 1/2 --> |x| will be less than 1

hence insufficient.

putting both together,

x < 0 and -1< x < 0 and x > 1
this limits x to -1< x < 0, thus |x| will be a positive fraction and will always be less than 1.

I second that... my slip.... i missed the greater part in ST 1...... ! Yes.. it should be C....
_________________

Cheers!
JT...........
If u like my post..... payback in Kudos!!

|For CR refer Powerscore CR Bible|For SC refer Manhattan SC Guide|

~~Better Burn Out... Than Fade Away~~

Director
Joined: 01 Aug 2008
Posts: 768
Followers: 4

Kudos [?]: 626 [1] , given: 99

Re: |x| less than 1? [#permalink]

### Show Tags

06 Jan 2010, 14:49
1
KUDOS
firasath, fantastic explanation. I missed taking fractions into account.

+1 for ya.
Math Expert
Joined: 02 Sep 2009
Posts: 36609
Followers: 7099

Kudos [?]: 93513 [1] , given: 10568

Re: |x| less than 1? [#permalink]

### Show Tags

06 Jan 2010, 14:54
1
KUDOS
Expert's post
1
This post was
BOOKMARKED
jeeteshsingh wrote:
ugimba wrote:
If x is not equal to 0, is |x| less than 1?

(1) $$x/|x| < x$$
(2) |x| > x

IMO A...

ST 1: can be written as $$x < x * |x|$$ since multiplying by |x| which is always positive would not change the sign.

This gives x as -1<x<0.... therefore sufficient...

ST2: |x| > x gives value of x as x<0... hence could be -2,-3.... Therefore not sufficient...

Cheers!
JT

If $$x$$ is not equal to $$0$$, is $$|x|$$ less than $$1$$?

Q: is $$-1<x<1$$ true?

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

$$x<0$$ --> $$\frac{x}{-x} < x$$ --> $$x>-1$$, but as $$x<0$$, then --> $$-1<x<0$$;

$$x>0$$ --> $$\frac{x}{x} < x$$ --> $$x>1$$.

So x can be as in the range {-1,1} as well as out of this range. Not sufficient.

(2) $$|x| > x$$ --> $$x<0$$. Not sufficient.

(1)+(2) Intersection of the ranges from (1) and (2) is: $$-1<x<0$$. Every $$x$$ from this range is in the range {-1,1}. Sufficient.
_________________
Manager
Joined: 24 Dec 2009
Posts: 224
Followers: 2

Kudos [?]: 39 [0], given: 3

Re: |x| less than 1? [#permalink]

### Show Tags

07 Jan 2010, 10:54

1. For Condition 1, x can either be positive integer or -ve fraction. Its true for both i.e if x = 2,3, etc
or x = -1/2, -1/3, etc.
Hence not sufficient.

2. For condition 2, X has to be -ve fraction or integer. Its true only when x = -1/2, -1/3, etc. or x=-1, -2, etc.
Hence B not sufficient.

3.Combine 1 & 2. True only when x is -ve fraction i.e. x=-1/2, -1/3, etc.
Hence |x| < 1.
Hence C should be the answer.
Manager
Joined: 06 Jan 2010
Posts: 73
Followers: 2

Kudos [?]: 13 [0], given: 15

Re: |x| less than 1? [#permalink]

### Show Tags

10 Jan 2010, 09:50
mod(x)/(x) can be 1 or -1
x ranges from (-1) to (infinity)
or (1) to infinity.

mod(x) > x means x is definitely a negative number

so x is between - 1 and 0 and is hence a fraction.
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 13510
Followers: 577

Kudos [?]: 163 [0], given: 0

Re: If x is not equal to 0, is |x| less than 1? [#permalink]

### Show Tags

06 Feb 2014, 02:32
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 13510
Followers: 577

Kudos [?]: 163 [0], given: 0

Re: If x is not equal to 0, is |x| less than 1? [#permalink]

### Show Tags

03 Oct 2015, 10:35
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Manager
Joined: 23 Sep 2015
Posts: 96
Concentration: General Management, Finance
GMAT 1: 680 Q46 V38
GMAT 2: 690 Q47 V38
GPA: 3.5
Followers: 0

Kudos [?]: 14 [0], given: 213

Re: If x is not equal to 0, is |x| less than 1? [#permalink]

### Show Tags

10 Nov 2015, 13:43
For this

(1) if x>0 we have 1<x
if x<0 we have -1<x

So NS

(2) if x>0 x>x --> Does not work
if x<0 -- > -x>x --> x<0

So NS

(1)+(2)

taking only the x<0 accounts for both of them, since X>0 in case 2 is not possible,

therefore we get the rank -1<x<0 and C is the answer.
Intern
Joined: 11 Jun 2015
Posts: 15
Followers: 0

Kudos [?]: 0 [0], given: 66

Re: If x is not equal to 0, is |x| less than 1? [#permalink]

### Show Tags

13 Nov 2015, 08:21
Bunuel wrote:
jeeteshsingh wrote:
ugimba wrote:
If x is not equal to 0, is |x| less than 1?

(1) $$x/|x| < x$$
(2) |x| > x

IMO A...

ST 1: can be written as $$x < x * |x|$$ since multiplying by |x| which is always positive would not change the sign.

This gives x as -1<x<0.... therefore sufficient...

ST2: |x| > x gives value of x as x<0... hence could be -2,-3.... Therefore not sufficient...

Cheers!
JT

If $$x$$ is not equal to $$0$$, is $$|x|$$ less than $$1$$?

Q: is $$-1<x<1$$ true?

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

$$x<0$$ --> $$\frac{x}{-x} < x$$ --> $$x>-1$$, but as $$x<0$$, then --> $$-1<x<0$$;

$$x>0$$ --> $$\frac{x}{x} < x$$ --> $$x>1$$.

So x can be as in the range {-1,1} as well as out of this range. Not sufficient.

(2) $$|x| > x$$ --> $$x<0$$. Not sufficient.

(1)+(2) Intersection of the ranges from (1) and (2) is: $$-1<x<0$$. Every $$x$$ from this range is in the range {-1,1}. Sufficient.

Hi Bunuel,
Thanks for this explanation. Bunuel, I am still having a hard time trying to understand the concepts behind the process followed to solve statement 1 in this question.

My reasoning is the following: for example if |x-3|=2, then x-3=2 and x-3=-2. If I follow this same reasoning then for statement one I get the following:

$$\frac{x}{|x|} < x$$

x/x<x ---> 1<x
x/x<-x ---> 1<-x ---> -1>x

What is the flaw in my reasoning?

Thanks so much for your help!
Math Forum Moderator
Joined: 20 Mar 2014
Posts: 2654
Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44
GPA: 3.7
WE: Engineering (Aerospace and Defense)
Followers: 117

Kudos [?]: 1342 [1] , given: 789

Re: If x is not equal to 0, is |x| less than 1? [#permalink]

### Show Tags

13 Nov 2015, 08:35
1
KUDOS
Expert's post
angierch wrote:

Hi Bunuel,
Thanks for this explanation. Bunuel, I am still having a hard time trying to understand the concepts behind the process followed to solve statement 1 in this question.

My reasoning is the following: for example if |x-3|=2, then x-3=2 and x-3=-2. If I follow this same reasoning then for statement one I get the following:

$$\frac{x}{|x|} < x$$

x/x<x ---> 1<x
x/x<-x ---> 1<-x ---> -1>x

What is the flaw in my reasoning?

Thanks so much for your help!

Firstly, |x|<1 ---> -1<x<1 . So the question asks whether x is between -1 and 1 with x$$\neq$$0

Your interpretation of |x-3|=2 is correct by taking the 2 cases. But the way you are approaching statement 1 is not correct. Look below.

Statement 1, x/|x|<x .

Take 2 cases.

Case 1: when x > 0 ---> |x| = x ---> x/|x|<x ---> x>1 . Now you had assumed that x>0 and it gave you x>1. Thus the range for x satisfying this case will thus become x>1. This will give a "no" to "is |x|<1 or is -1<x<1?".

case 2: when x < 0 ---> |x| = -x ---> x/|x|<x ---> x>-1 . Now you had assumed that x<0 and it gave you x>-1. Thus the range for x satisfying this case will thus become -1<x<0. This will give a "yes" to "is |x|<1 or is -1<x<1?".

Thus you get 2 different cases for the same statement, making statement 1 not sufficient.

The issue with your solution is that you are not taking the intersection of the possible values as I have shown in 2 cases above.

|x-3| = 2 --> x-3=2 ONLY when x-3 $$\geq$$0 .

Similarly, |x-3| = 2 --> x-3=-2 ONLY when x-3 < 0.

hope this helps.
_________________

Thursday with Ron updated list as of July 1st, 2015: http://gmatclub.com/forum/consolidated-thursday-with-ron-list-for-all-the-sections-201006.html#p1544515
Inequalities tips: http://gmatclub.com/forum/inequalities-tips-and-hints-175001.html
Debrief, 650 to 750: http://gmatclub.com/forum/650-to-750-a-10-month-journey-to-the-score-203190.html

Intern
Joined: 11 Jun 2015
Posts: 15
Followers: 0

Kudos [?]: 0 [0], given: 66

Re: If x is not equal to 0, is |x| less than 1? [#permalink]

### Show Tags

15 Nov 2015, 19:14
Engr2012 wrote:
angierch wrote:

Hi Bunuel,
Thanks for this explanation. Bunuel, I am still having a hard time trying to understand the concepts behind the process followed to solve statement 1 in this question.

My reasoning is the following: for example if |x-3|=2, then x-3=2 and x-3=-2. If I follow this same reasoning then for statement one I get the following:

$$\frac{x}{|x|} < x$$

x/x<x ---> 1<x
x/x<-x ---> 1<-x ---> -1>x

What is the flaw in my reasoning?

Thanks so much for your help!

Firstly, |x|<1 ---> -1<x<1 . So the question asks whether x is between -1 and 1 with x$$\neq$$0

Your interpretation of |x-3|=2 is correct by taking the 2 cases. But the way you are approaching statement 1 is not correct. Look below.

Statement 1, x/|x|<x .

Take 2 cases.

Case 1: when x > 0 ---> |x| = x ---> x/|x|<x ---> x>1 . Now you had assumed that x>0 and it gave you x>1. Thus the range for x satisfying this case will thus become x>1. This will give a "no" to "is |x|<1 or is -1<x<1?".

case 2: when x < 0 ---> |x| = -x ---> x/|x|<x ---> x>-1 . Now you had assumed that x<0 and it gave you x>-1. Thus the range for x satisfying this case will thus become -1<x<0. This will give a "yes" to "is |x|<1 or is -1<x<1?".

Thus you get 2 different cases for the same statement, making statement 1 not sufficient.

The issue with your solution is that you are not taking the intersection of the possible values as I have shown in 2 cases above.

|x-3| = 2 --> x-3=2 ONLY when x-3 $$\geq$$0 .

Similarly, |x-3| = 2 --> x-3=-2 ONLY when x-3 < 0.

hope this helps.

Thank you very much!
Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 2623
GPA: 3.82
Followers: 174

Kudos [?]: 1456 [0], given: 0

Re: If x is not equal to 0, is |x| less than 1? [#permalink]

### Show Tags

17 Nov 2015, 09:24
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

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

(1) x/|x| < x
(2) |x| > x

If the range of the condition falls into that of the condition in terms of inequalities, the condition is sufficient.

There is 1 variable in the original condition, and there are 2 equations provided by the 2 conditions, so there is high chance (D) will be our answer.
For condition 1, if x>0, x/|x|<x --> x/x<x --> 1<x, then 1<x
if x<0, x/|x|<x --> x/-x<x --> -1<x, then -1<x<0. Therefore this condition is insufficient because this range does not fall into that of the question.
For condition 2, |x|>x --> x<0. This is insufficient for the same reason.
Looking at them together, however, -1<x<0 falls into the range of the question, so this is sufficient, and the answer becomes (C).

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
Find a 10% off coupon code for GMAT Club members.
“Receive 5 Math Questions & Solutions Daily”
Unlimited Access to over 120 free video lessons - try it yourself

Re: If x is not equal to 0, is |x| less than 1?   [#permalink] 17 Nov 2015, 09:24
Similar topics Replies Last post
Similar
Topics:
If x is not equal to 0, is |x| less than 1? 3 01 Jan 2015, 18:05
24 If x is not equal to 0, is |x| less than 1? 14 06 May 2010, 03:20
182 If x is not equal to 0, is |x| less than 1? (1) x/|x|< x 93 01 Nov 2009, 08:25
1 If x is not equal to 0, is |x| less than 1? (1) x/|x| < x 3 10 Sep 2009, 03:32
9 If x is not equal to 0, is |x| less than 1? (1) x/|x|< x 10 07 Aug 2009, 14:19
Display posts from previous: Sort by