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

 It is currently 05 Dec 2016, 13:40

### 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 zero, is |x| < 1 ?

Author Message
TAGS:

### Hide Tags

Moderator
Joined: 01 Sep 2010
Posts: 3032
Followers: 768

Kudos [?]: 6337 [0], given: 991

If x is not equal to zero, is |x| < 1 ? [#permalink]

### Show Tags

30 Sep 2012, 05:20
7
This post was
BOOKMARKED
00:00

Difficulty:

45% (medium)

Question Stats:

61% (02:08) correct 39% (01:16) wrong based on 299 sessions

### HideShow timer Statistics

If x is not equal to zero, is |x| < 1 ?

(1) x^2 < 1

(2) |x| < 1/x

[Reveal] Spoiler:
I would like to know if I do in the right manner

question: x id non zero, is $$| x | < 1$$ ??? now to be < 1 x must be negative, so the question become " is x negative ???

1) for a square to be < 1 the must be for example $$\frac{- 1}{2}$$ ----> x is negative . suff

2) for x to be minor of a certain number that is the reciprocal i.e. : $$-2 < - 1/2$$ --------> x is negative. suff

is correct or I should rethink the entire part regarding inequalities ?? because this statement should not take you more that 50 seconds to solve. otherwise you get in trouble with this exam.

Thanks
[Reveal] Spoiler: OA

_________________

Last edited by Bunuel on 01 Oct 2012, 04:50, edited 1 time in total.
Renamed the topic and edited the question.
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7066
Location: Pune, India
Followers: 2084

Kudos [?]: 13277 [4] , given: 221

Re: If x is not equal to zero, is | x | < 1 ?? [#permalink]

### Show Tags

30 Sep 2012, 21:50
4
KUDOS
Expert's post
carcass wrote:
If x is not equal to zero, is $$| x | < 1$$ ??

1) $$x^2$$ $$< 1$$

2) $$| x |$$ < $$\frac{1}{x}$$

I would like to know if I do in the right manner

question: x id non zero, is $$| x | < 1$$ ??? now to be < 1 x must be negative, so the question become " is x negative ???

1) for a square to be < 1 the must be for example $$\frac{- 1}{2}$$ ----> x is negative . suff

2) for x to be minor of a certain number that is the reciprocal i.e. : $$-2 < - 1/2$$ --------> x is negative. suff

is correct or I should rethink the entire part regarding inequalities ?? because this statement should not take you more that 50 seconds to solve. otherwise you get in trouble with this exam.

Thanks

I would suggest you to keep 4 ranges in mind when dealing with squares, reciprocals etc.

............... -1........ 0 ........ 1 ..............

Less than -1, -1 to 0, 0 to 1 and greater than 1

You can do the question logically or by plugging in numbers.

Ques: If x is not equal to zero, is $$| x | < 1$$ ??
Logically, | x | implies distance from 0 on the number line. So the question becomes "Is x a distance of less than 1 away from 0?" i.e. "Is -1 < x < 1?" given x is not 0.

Statement 1: $$x^2$$ $$< 1$$
Square of a number will be less than 1 only if the absolute value of the number is less than 1. This means $$| x | < 1$$. The number needn't be negative. If it is positive, it should be less than 1. If it negative, it should be greater than -1.
Or, do you remember how to solve inequalities using the wave? $$x^2 < 1$$ is $$x^2 - 1 < 0$$ which is $$(x - 1)(x + 1) < 0$$. This implies -1 < x < 1 but x is not 0.
Or plug in numbers from the given 4 ranges. YOu will see that x must lie in -1 < x < 1.
Sufficient.

Statement 2: $$| x | < \frac{1}{x}$$
First of all, x cannot be negative since | x | is never negative. Since | x | is less than 1/x, 1/x must be positive. Also, x cannot be greater than 1 since then, | x | will be greater than 1/x.
Or plug in numbers from the given 4 ranges. You will see that x must lie in 0 < x < 1.
Sufficient.

_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for \$199

Veritas Prep Reviews

Math Expert
Joined: 02 Sep 2009
Posts: 35867
Followers: 6840

Kudos [?]: 89958 [3] , given: 10384

Re: If x is not equal to zero, is |x| < 1 ? [#permalink]

### Show Tags

01 Oct 2012, 05:05
3
KUDOS
Expert's post
1
This post was
BOOKMARKED
If x is not equal to zero, is |x| < 1 ?

Is $$|x| < 1$$? --> is $$-1<x<1$$ ($$x\neq{0}$$).

(1) x^2 < 1 --> -1<x<1. Directly answers the question. Sufficient.

(2) |x| < 1/x. Since the left hand side of the inequity (|x|) is an absolute value, which cannot be negative (actually in this case we know that its positive as we are given that $$x\neq{0}$$) then the right hand side (1/x) must also be positive, which means that $$x>0$$, so $$|x|=x$$.

Hence, $$|x| < \frac{1}{x}$$ becomes $$x<\frac{1}{x}$$. Since we know that $$x>0$$ we can safely cross-multiply: $$x^2<1$$. The same inequality as in (1). Sufficient.

_________________
Director
Joined: 22 Mar 2011
Posts: 612
WE: Science (Education)
Followers: 98

Kudos [?]: 866 [2] , given: 43

Re: If x is not equal to zero, is | x | < 1 ?? [#permalink]

### Show Tags

30 Sep 2012, 05:34
2
KUDOS
1
This post was
BOOKMARKED
carcass wrote:
If x is not equal to zero, is $$| x | < 1$$ ??

1) $$x^2$$ $$< 1$$

2) $$| x |$$ < $$\frac{1}{x}$$

I would like to know if I do in the right manner

question: x id non zero, is $$| x | < 1$$ ??? now to be < 1 x must be negative, so the question become " is x negative ???

1) for a square to be < 1 the must be for example $$\frac{- 1}{2}$$ ----> x is negative . suff

2) for x to be minor of a certain number that is the reciprocal i.e. : $$-2 < - 1/2$$ --------> x is negative. suff

is correct or I should rethink the entire part regarding inequalities ?? because this statement should not take you more that 50 seconds to solve. otherwise you get in trouble with this exam.

Thanks

The condition $$x$$ non-zero was given just because statement (2) has $$x$$ in the denominator.

x id non zero, is $$| x | < 1$$ ???
Not necessarily. $$x$$ can be greater than $$1$$ or less than $$-1$$, in which case, definitely $$|x|$$ is not less than $$1.$$

now to be < 1 x must be negative, so the question become " is x negative ??? NO
$$x$$ can be between $$0$$ and $$1$$.

(1) $$x^2$$ $$< 1$$, take square root from both sides and obtain $$|x|<1$$, because $$\sqrt{x^2}=|x|$$.
Sufficient.

(2) Since $$|x|<\frac{1}{x}$$, it follows that $$x>0$$, because $$|x|>0$$. $$x$$ cannot be negative!!!
Then, the given inequality becomes $$x<\frac{1}{x}$$, or $$x^2<1$$ (we can multiply both sides by $$x$$, which is positive) and we are again in the situation from (1) above.
Sufficient.

_________________

PhD in Applied Mathematics
Love GMAT Quant questions and running.

Manager
Status: Fighting again to Kill the GMAT devil
Joined: 02 Jun 2009
Posts: 137
Location: New Delhi
WE 1: Oil and Gas - Engineering & Construction
Followers: 1

Kudos [?]: 55 [0], given: 48

Re: If x is not equal to zero, is | x | < 1 ?? [#permalink]

### Show Tags

30 Sep 2012, 06:58
Perfect Explanation, Evajager, @Carcass this is a very common mistake which I am also prone to make.
Thanks @Evajager - Your explanation for {B}
Quote:
(2) Since , it follows that , because . cannot be negative!!!
Then, the given inequality becomes , or (we can multiply both sides by , which is positive) and we are again in the situation from (1) above.
Sufficient.

I remember skipping a Question just like this in one of my exams because I gt stuck!!! in solving - | x | < 1/x

Thanks guys you rock!!!
_________________

Giving Kudos, is a great Way to Help the GC Community Kudos

Director
Joined: 22 Mar 2011
Posts: 612
WE: Science (Education)
Followers: 98

Kudos [?]: 866 [0], given: 43

Re: If x is not equal to zero, is | x | < 1 ?? [#permalink]

### Show Tags

30 Sep 2012, 12:29
EvaJager wrote:
carcass wrote:
If x is not equal to zero, is $$| x | < 1$$ ??

1) $$x^2$$ $$< 1$$

2) $$| x |$$ < $$\frac{1}{x}$$

I would like to know if I do in the right manner

question: x id non zero, is $$| x | < 1$$ ??? now to be < 1 x must be negative, so the question become " is x negative ???

1) for a square to be < 1 the must be for example $$\frac{- 1}{2}$$ ----> x is negative . suff

2) for x to be minor of a certain number that is the reciprocal i.e. : $$-2 < - 1/2$$ --------> x is negative. suff

is correct or I should rethink the entire part regarding inequalities ?? because this statement should not take you more that 50 seconds to solve. otherwise you get in trouble with this exam.

Thanks

The condition $$x$$ non-zero was given just because statement (2) has $$x$$ in the denominator.

x id non zero, is $$| x | < 1$$ ???
Not necessarily. $$x$$ can be greater than $$1$$ or less than $$-1$$, in which case, definitely $$|x|$$ is not less than $$1.$$

now to be < 1 x must be negative, so the question become " is x negative ??? NO
$$x$$ can be between $$0$$ and $$1$$.

(1) $$x^2$$ $$< 1$$, take square root from both sides and obtain $$|x|<1$$, because $$\sqrt{x^2}=|x|$$.
Sufficient.

(2) Since $$|x|<\frac{1}{x}$$, it follows that $$x>0$$, because $$|x|>0$$. $$x$$ cannot be negative!!!
Then, the given inequality becomes $$x<\frac{1}{x}$$, or $$x^2<1$$ (we can multiply both sides by $$x$$, which is positive) and we are again in the situation from (1) above.
Sufficient.

Correction: in (2) and we are again in the situation from (1) above. - not exactly, but similar as we have $$x^2<1$$ but in addition $$x>0.$$ In (1) $$x$$ could also be negative.
The conclusion is still correct, as now $$|x|=x<1.$$
_________________

PhD in Applied Mathematics
Love GMAT Quant questions and running.

Moderator
Joined: 01 Sep 2010
Posts: 3032
Followers: 768

Kudos [?]: 6337 [0], given: 991

Re: If x is not equal to zero, is | x | < 1 ?? [#permalink]

### Show Tags

01 Oct 2012, 03:24
VeritasPrepKarishma wrote:
carcass wrote:
If x is not equal to zero, is $$| x | < 1$$ ??

1) $$x^2$$ $$< 1$$

2) $$| x |$$ < $$\frac{1}{x}$$

I would like to know if I do in the right manner

question: x id non zero, is $$| x | < 1$$ ??? now to be < 1 x must be negative, so the question become " is x negative ???

1) for a square to be < 1 the must be for example $$\frac{- 1}{2}$$ ----> x is negative . suff

2) for x to be minor of a certain number that is the reciprocal i.e. : $$-2 < - 1/2$$ --------> x is negative. suff

is correct or I should rethink the entire part regarding inequalities ?? because this statement should not take you more that 50 seconds to solve. otherwise you get in trouble with this exam.

Thanks

I would suggest you to keep 4 ranges in mind when dealing with squares, reciprocals etc.

............... -1........ 0 ........ 1 ..............

Less than -1, -1 to 0, 0 to 1 and greater than 1

You can do the question logically or by plugging in numbers.

Ques: If x is not equal to zero, is $$| x | < 1$$ ??
Logically, | x | implies distance from 0 on the number line. So the question becomes "Is x a distance of less than 1 away from 0?" i.e. "Is -1 < x < 1?" given x is not 0.

Statement 1: $$x^2$$ $$< 1$$
Square of a number will be less than 1 only if the absolute value of the number is less than 1. This means $$| x | < 1$$. The number needn't be negative. If it is positive, it should be less than 1. If it negative, it should be greater than -1.
Or, do you remember how to solve inequalities using the wave? $$x^2 < 1$$ is $$x^2 - 1 < 0$$ which is $$(x - 1)(x + 1) < 0$$. This implies -1 < x < 1 but x is not 0.
Or plug in numbers from the given 4 ranges. YOu will see that x must lie in -1 < x < 1.
Sufficient.

Statement 2: $$| x | < \frac{1}{x}$$
First of all, x cannot be negative since | x | is never negative. Since | x | is less than 1/x, 1/x must be positive. Also, x cannot be greater than 1 since then, | x | will be greater than 1/x.
Or plug in numbers from the given 4 ranges. You will see that x must lie in 0 < x < 1.
Sufficient.

This is the same reasoning that I followed in the first instance with some errors but the path was correct. Specifically the stimulus evaluation : $$| x | < 1$$ ------ in this scenario the first one is $$x < 1$$ AND $$-x < 1$$ so $$x > -1$$ ------> $$- 1 < x < 1$$

Thanks Mod
_________________
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 12876
Followers: 559

Kudos [?]: 158 [0], given: 0

Re: If x is not equal to zero, is |x| < 1 ? [#permalink]

### Show Tags

22 Jul 2014, 03:52
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: 12876
Followers: 559

Kudos [?]: 158 [0], given: 0

Re: If x is not equal to zero, is |x| < 1 ? [#permalink]

### Show Tags

27 Oct 2015, 04:56
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: 12876
Followers: 559

Kudos [?]: 158 [0], given: 0

Re: If x is not equal to zero, is |x| < 1 ? [#permalink]

### Show Tags

03 Nov 2016, 14:33
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.
_________________
Intern
Joined: 09 Aug 2016
Posts: 35
Followers: 0

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

If x is not equal to zero, is |x| < 1 ? [#permalink]

### Show Tags

20 Nov 2016, 14:48
The general rule of thumb with DS + ineq questions is if you can avoid plug numbers DO IT!

If x is not equal to zero, is |x| < 1 ?

Unraveling the question we have case1 where x < 1 OR case 2 where x > -1

REMEMBER that with ineq. DS questions A suffiecient asnswer is the answer that verifies case1 OR case2 OR BOTH

(1) x^2 < 1
"ROOTING" both sides (i.e. raised both sides in 1/2 power) we have sqrt(x^2) = sqrt(1) = 1 hence |x| < 1 which is SUFFICIENT as verifies case1 and case2 (effectivly if you solve it you get exactly the 2 cases from the question stem). Is important to know that "ROOTING" is allowed here because BOTH sides of the enequality are NOT NEGATIVE.

(2) |x| < 1/x
Solving the ineq. above gives two cases:

caseA: x < 1 / x and caseB x > - 1/x

CaseA: Since x>0 we can multiply both sides and get x^2 < 1. Rooting both sides again (as shown in (1) above) you get |x| < 1 this verifies again case1 and case2 from the question stem. At this point YOU DONT need to go and check caseB.

If x is not equal to zero, is |x| < 1 ?   [#permalink] 20 Nov 2016, 14:48
Similar topics Replies Last post
Similar
Topics:
1 If 0 < x < 1, is the hundredths digit of x equal to 9 ? 1 09 Jun 2015, 11:30
x| < 1? 1) |x+1| = 2|x-1| 2) |x-3| does not equal 0 4 01 May 2011, 19:59
180 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