Find all School-related info fast with the new School-Specific MBA Forum

It is currently 27 Aug 2016, 22:22
GMAT Club Tests

Close

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
Your Progress

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

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

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

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Manager
Manager
avatar
Joined: 07 May 2013
Posts: 109
Followers: 0

Kudos [?]: 21 [0], given: 1

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

Show Tags

New post 02 Jun 2014, 21:11
Buneul, you are saying that x>-1, suppose we substitute 0 in the inequality we get 0<0. How is this true????
Expert Post
Math Expert
User avatar
Joined: 02 Sep 2009
Posts: 34457
Followers: 6280

Kudos [?]: 79691 [0], given: 10022

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

Show Tags

New post 03 Jun 2014, 01:18
madn800 wrote:
Buneul, you are saying that x>-1, suppose we substitute 0 in the inequality we get 0<0. How is this true????


First of all please read the whole thread. For example, check the following posts:
x-x-x-which-of-the-following-must-be-true-about-x-13943-40.html#p772618
x-x-x-which-of-the-following-must-be-true-about-x-13943-40.html#p773277
x-x-x-which-of-the-following-must-be-true-about-x-13943-40.html#p807569

x=0 does not satisfy x/|x| < x, so x cannot be 0.

x/|x| < x, means that -1<x<0 or x>1. ANY x from these ranges will be greater than -1.
_________________

New to the Math Forum?
Please read this: All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

GMAT Club Legend
GMAT Club Legend
User avatar
Joined: 09 Sep 2013
Posts: 11106
Followers: 511

Kudos [?]: 134 [0], given: 0

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

Show Tags

New post 02 Aug 2015, 11:06
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 Books | GMAT Club Tests | Best Prices on GMAT Courses | GMAT Mobile App | Math Resources | Verbal Resources

Intern
Intern
avatar
Joined: 03 Jul 2015
Posts: 40
Followers: 0

Kudos [?]: 10 [0], given: 27

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

Show Tags

New post 31 Aug 2015, 10:24
Bunuel wrote:
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 are given that \(\frac{x}{|x|}<x\) (this is a true inequality), so first of all we should find the ranges of \(x\) for which this inequality holds true.

\(\frac{x}{|x|}< x\) multiply both sides of inequality by \(|x|\) (side note: we can safely do this as absolute value is non-negative and in this case we know it's not zero too) --> \(x<x|x|\) --> \(x(|x|-1)>0\):
Either \(x>0\) and \(|x|-1>0\), so \(x>1\) or \(x<-1\) --> \(x>1\);
Or \(x<0\) and \(|x|-1<0\), so \(-1<x<1\) --> \(-1<x<0\).

So we have that: \(-1<x<0\) or \(x>1\). Note \(x\) is ONLY from these ranges.

Option B says: \(x>-1\) --> ANY \(x\) from above two ranges would be more than -1, so B is always true.

Answer: B.

nonameee wrote:
Quote:
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



Bunuel, I think there's a mistake in the question or in the answer choices:

Here's my solution:
1) x<0:
-x/x < x
-1<x<0

2) x>=0
x/x<x
x>1

The solution of the inequality is then:
-1<x<0 union x>1

The answer can't be B, since let x=1/2. We get:
(1/2)/(1/2) < (1/2)
1< 1/2
Contradiction.

I think the answer should be A since it satisfies all the scenarios.

Can you please clarify?


The options are not supposed to be the solutions of inequality \(\frac{x}{|x|}<x\).

Hope it's clear.


how did you came up −1<x<0 or x>1. i can not understand this. i am sorry if i so silly for you guys. but plz help me
Intern
Intern
avatar
Joined: 03 Jul 2015
Posts: 40
Followers: 0

Kudos [?]: 10 [0], given: 27

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

Show Tags

New post 31 Aug 2015, 10:38
Bunuel wrote:
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 are given that \(\frac{x}{|x|}<x\) (this is a true inequality), so first of all we should find the ranges of \(x\) for which this inequality holds true.

\(\frac{x}{|x|}< x\) multiply both sides of inequality by \(|x|\) (side note: we can safely do this as absolute value is non-negative and in this case we know it's not zero too) --> \(x<x|x|\) --> \(x(|x|-1)>0\):
Either \(x>0\) and \(|x|-1>0\), so \(x>1\) or \(x<-1\) --> \(x>1\);
Or \(x<0\) and \(|x|-1<0\), so \(-1<x<1\) --> \(-1<x<0\).

So we have that: \(-1<x<0\) or \(x>1\). Note \(x\) is ONLY from these ranges.

Option B says: \(x>-1\) --> ANY \(x\) from above two ranges would be more than -1, so B is always true.

Answer: B.

nonameee wrote:
Quote:
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



Bunuel, I think there's a mistake in the question or in the answer choices:

Here's my solution:
1) x<0:
-x/x < x
-1<x<0

2) x>=0
x/x<x
x>1

The solution of the inequality is then:
-1<x<0 union x>1

The answer can't be B, since let x=1/2. We get:
(1/2)/(1/2) < (1/2)
1< 1/2
Contradiction.

I think the answer should be A since it satisfies all the scenarios.

Can you please clarify?


The options are not supposed to be the solutions of inequality \(\frac{x}{|x|}<x\).

Hope it's clear.

x<x|x| --> x(|x|−1)>0:
Either x>0 and |x|−1>0, so x>1 or x<−1 --> x>1;
Or x<0 and |x|−1<0, so −1<x<1 --> −1<x<0.
i can not undersdant the coloured portion from where we can derive this?
1 KUDOS received
Intern
Intern
avatar
Joined: 03 Jul 2015
Posts: 40
Followers: 0

Kudos [?]: 10 [1] , given: 27

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

Show Tags

New post 31 Aug 2015, 11:14
1
This post received
KUDOS
fluke wrote:
syh244 wrote:
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

Answer is A. I hope the following explanation permanently sets A as the right answer for this problem. I see that some people tried simplifying the formula first, and some people performed a common mistake while doing so. Lets look at what many people tried to do.

Step 1: multiply |x| to both sides, resulting in x<x*|x| (this step is correct as |x| will always be positive and thus multiplying |x| to both sides will not cause ambiguity in the direction of the inequality)
Step 2: divide x to both sides, resulting in 1<|x| (this step is incorrect as x could be positive or negative, and such ambiguity restricts such operation. very common mistake that should be avoided at all costs as GMAT question makers will use this against you)

So.... lets go back to Step 1, x<x*|x|
From here, you can further simplify the equation to 0<x(|x|-1) as many people above did, but simplifying the equation this will only create more brain damage than simplify the problem.

The best course of action after Step 1 is to plug in numbers (or even plug in numbers straight to the original equation without performing Step 1).

-2 does not work because -2<-2*|-2| = -2<-4, which is not true
-1 does not work because -1<-1*|-1| = -1<-1, which is not true
-1/2 works because -1/2<-1/2*|-1/2| = -1/2<-1/4, which is true
0 does not work because 0<0*|0| = 0<0, which is not true
1/2 does not work because 1/2<1/2*|1/2| = 1/2<1/4, which is not true
1 does not work because 1<1*|1| = 1<1, which is not true
2 works because 2 < 2*|2| = 2<4, which is true
3 works because 3<3|3 = 3<9, which is true

So we have a situation in which 0>x>-1, and x>1.

Now here is where many people are getting REALLY confused.
Many people say that the right answer is B because, yes, x>-1, BUT it has certain restrictions. The statement x>-1 should also include 0 and 1/2 as the right solutions, but the original equation fails when these numbers are plugged in, making x>1 the only right answer for the equation. Yes, I understand that the answer A disregards the portion where 0>x>-1, but who cares, the question is asking "which of the following must be true about x" and not "what are the solutions for x"


Your analysis is amazing, but the summing up is not entirely correct.

As you yourself said:
the question is asking "which of the following must be true about x" and not "what are the solutions for x".

When the condition is given:
x/|x|<x

And we get the range for x:
-1<x<0 OR x>1.

It is imperative that "x" MUST BE in that range. So, x just can't be less than equal to -1, OR Anything between 0 AND 1, inclusive, such as "1/2".

If there is a value of x, it must be in the range specified above.

So,
(A) x>1: It is one of the possible ranges. But, we don't know what will x bear. What if x=-0.5. Not necessarily true.
(B) x>-1: No matter which value "x" bears, it will always be >-1. Must be true.
(C) |x|<1: This CAN be true, but what if x = 2. Not necessarily true.
(D) |x|=1: This cannot be true. Both -1 AND +1 are outside the range.
(E) |x|^2>1: x<-1 OR x>1. What if x=-0.5. Not necessarily true.

Ans: "B"

hi bunnel, i can not understand how did you make the range
Intern
Intern
avatar
Joined: 29 May 2015
Posts: 8
Followers: 0

Kudos [?]: 0 [0], given: 4

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

Show Tags

New post 15 Sep 2015, 01:18
I hope this helps.


From x/|x|<x, we know that :

1. x cannot be 0, since we can't divide a number by 0
2. x cannot be greater than 1, since we are dividing the number by the same number
3. x cannot be less than -1, since we are dividing the number by then same number

So we get, -1<x<1
From here, if we want x/|x|<x to be true, pick some numbers and you will see that only negative fraction works.
So it means,
-1<x<0

a) x>1 --> false
b) x>-1 --> correct
c) |x|<1 -->false, since x can be positive fraction
d) |x|=1 --> false, since x cannot be greater than 0
e) |x|^2>1 --> false, since squaring fraction gives you less than 0
Intern
Intern
avatar
Joined: 31 Oct 2015
Posts: 37
Followers: 0

Kudos [?]: 2 [0], given: 53

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

Show Tags

New post 27 Dec 2015, 12:48
Condition 1:

x > 0:

x/|x| = 1

x/|x| < x

1 < x

× -> (1, infinity]

Condition 2:

x < 0:

x/|x| = -1

-1 < x and x < 0


x -> (-1,0)

Conclusion: x -> (-1, 0) or (1, infinity]

A) x > 1 excludes values of x (-1, 0)
B) x > -1 includes all possible values of x. CORRECT
C) |x| < 1 covers (-1, 0) but not (1, infinity]
D) |x| = 1 incorrect as |x| != 1 in any circumstance.
E) |x|^2 >1 fails to cover values of x -> (-1, 0)
Director
Director
User avatar
Joined: 12 Aug 2015
Posts: 974
Followers: 17

Kudos [?]: 132 [0], given: 267

GMAT ToolKit User CAT Tests
Re: x/|x|<x. which of the following must be true about x ? [#permalink]

Show Tags

New post 10 Mar 2016, 11:07
Excellent Question Here we just need to dilute the condition with remarks that x>0 => |x|=x and if x<0 => |x|=-x
_________________

Give me a hell yeah ...!!!!!

Manager
Manager
avatar
Joined: 13 Apr 2016
Posts: 55
WE: Operations (Hospitality and Tourism)
Followers: 0

Kudos [?]: 15 [0], given: 5

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

Show Tags

New post 19 May 2016, 05:20
Bunuel wrote:
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 are given that \(\frac{x}{|x|}<x\) (this is a true inequality), so first of all we should find the ranges of \(x\) for which this inequality holds true.

\(\frac{x}{|x|}< x\) multiply both sides of inequality by \(|x|\) (side note: we can safely do this as absolute value is non-negative and in this case we know it's not zero too) --> \(x<x|x|\) --> \(x(|x|-1)>0\):
Either \(x>0\) and \(|x|-1>0\), so \(x>1\) or \(x<-1\) --> \(x>1\);
Or \(x<0\) and \(|x|-1<0\), so \(-1<x<1\) --> \(-1<x<0\).

So we have that: \(-1<x<0\) or \(x>1\). Note \(x\) is ONLY from these ranges.

Option B says: \(x>-1\) --> ANY \(x\) from above two ranges would be more than -1, so B is always true.

Answer: B.

nonameee wrote:
Quote:
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



Bunuel, I think there's a mistake in the question or in the answer choices:

Here's my solution:
1) x<0:
-x/x < x
-1<x<0

2) x>=0
x/x<x
x>1

The solution of the inequality is then:
-1<x<0 union x>1

The answer can't be B, since let x=1/2. We get:
(1/2)/(1/2) < (1/2)
1< 1/2
Contradiction.

I think the answer should be A since it satisfies all the scenarios.

Can you please clarify?


The options are not supposed to be the solutions of inequality \(\frac{x}{|x|}<x\).

Hope it's clear.


if answer to this question is B then why not x=1 satisfy the situation.
Expert Post
Math Expert
User avatar
Joined: 02 Sep 2009
Posts: 34457
Followers: 6280

Kudos [?]: 79691 [0], given: 10022

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

Show Tags

New post 19 May 2016, 07:32
Intern
Intern
avatar
Joined: 13 Sep 2015
Posts: 29
Followers: 0

Kudos [?]: 0 [0], given: 28

GMAT ToolKit User
Re: x/|x|<x. which of the following must be true about x ? [#permalink]

Show Tags

New post 26 May 2016, 20:10
X>-1 includes the case x=0 ...so x>1
Re: x/|x|<x. which of the following must be true about x ?   [#permalink] 26 May 2016, 20:10

Go to page   Previous    1   2   3   [ 52 posts ] 

    Similar topics Author Replies Last post
Similar
Topics:
3 Experts publish their posts in the topic Which of the following is true about 0<|x|-4x<5? MathRevolution 4 26 Apr 2016, 06:33
4 If √x=x , then which of the following must be true ? amitabc 3 26 Oct 2014, 04:35
17 Experts publish their posts in the topic If |x|=−x, which of the following must be true? Mountain14 5 22 Mar 2014, 02:31
26 Experts publish their posts in the topic If x/|x|, which of the following must be true for all praveenvino 14 15 Jan 2011, 12:44
2 Experts publish their posts in the topic If x/|x| < x, which of the following must be true about tkarthi4u 25 06 Sep 2009, 22:14
Display posts from previous: Sort by

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

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  


GMAT Club MBA Forum Home| About| Terms and Conditions| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group and phpBB SEO

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.