GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 12 Dec 2018, 15:41

### 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

## Events & Promotions

###### Events & Promotions in December
PrevNext
SuMoTuWeThFrSa
2526272829301
2345678
9101112131415
16171819202122
23242526272829
303112345
Open Detailed Calendar
• ### The winning strategy for 700+ on the GMAT

December 13, 2018

December 13, 2018

08:00 AM PST

09:00 AM PST

What people who reach the high 700's do differently? We're going to share insights, tips and strategies from data we collected on over 50,000 students who used examPAL.
• ### GMATbuster's Weekly GMAT Quant Quiz, Tomorrow, Saturday at 9 AM PST

December 14, 2018

December 14, 2018

09:00 AM PST

10:00 AM PST

10 Questions will be posted on the forum and we will post a reply in this Topic with a link to each question. There are prizes for the winners.

# If x/|x|, which of the following must be true for all

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

### Hide Tags

Intern
Joined: 06 Nov 2010
Posts: 21
If x/|x|, which of the following must be true for all  [#permalink]

### Show Tags

Updated on: 09 Jul 2013, 08:56
1
23
00:00

Difficulty:

55% (hard)

Question Stats:

60% (01:38) correct 40% (01:59) wrong based on 769 sessions

### HideShow timer Statistics

If $$\frac{x}{|x|} \lt x$$, which of the following must be true about $$x$$? ($$x \ne 0$$)

A. $$x\gt 2$$
B. $$x \in (-1,0) \cup (1,\infty)$$
C. $$|x| \lt 1$$
D. $$|x| = 1$$
E. $$|x|^2 \gt 1$$

M24

Originally posted by praveenvino on 15 Jan 2011, 11:44.
Last edited by Bunuel on 09 Jul 2013, 08:56, edited 1 time in total.
Renamed the topic and edited the question.
##### Most Helpful Expert Reply
Math Expert
Joined: 02 Sep 2009
Posts: 51121
Re: range of root - GMAT Club test - M24  [#permalink]

### Show Tags

15 Jan 2011, 13:47
2
9
praveenvino wrote:
X/|X| < X . Which of the following must be true for all ?

a. X > 1
b. X is an element in (-1,0) U (1,inf)
c. |X| < 1
d. |X| = 1
e. |X|^2 > 1

Can some one explain how X can be zero for the above condition?

x is in the denominator so it can not equal to zero as division be zero is undefined.

Correct form of this question is below (m09 q22, discussed here: m09-q22-69937.html):

If $$\frac{x}{|x|} \lt x$$, which of the following must be true about $$x$$? ($$x \ne 0$$)
A. $$x\gt 2$$
B. $$x \in (-1,0) \cup (1,\infty)$$
C. $$|x| \lt 1$$
D. $$|x| = 1$$
E. $$|x|^2 \gt 1$$

$$\frac{x}{|x|}< x$$
Two cases:
A. $$x<0$$ --> $$\frac{x}{-x}<x$$ --> $$-1<x$$. But as we consider the range $$x<0$$ then $$-1<x<0$$

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

So the given inequality holds true in two ranges $$-1<x<0$$ and $$x>1$$.

Answer: B.

For more check: math-absolute-value-modulus-86462.html

Hope it helps.
_________________
##### General Discussion
Intern
Joined: 06 Nov 2010
Posts: 21
Re: range of root - GMAT Club test - M24  [#permalink]

### Show Tags

15 Jan 2011, 14:03
Thanks Bunuel. X not equals zero condition was actually missing in the question in m24. Thanks for your help.
Intern
Joined: 07 Oct 2014
Posts: 3
Re: If x/|x|, which of the following must be true for all  [#permalink]

### Show Tags

11 Dec 2014, 23:00
A must be true too.
If x>1 satisfy x/|x|<x
then x>2 will do too.
can anyone explain choice A? thanks!
Math Expert
Joined: 02 Sep 2009
Posts: 51121
Re: If x/|x|, which of the following must be true for all  [#permalink]

### Show Tags

12 Dec 2014, 05:51
pinguuu wrote:
A must be true too.
If x>1 satisfy x/|x|<x
then x>2 will do too.
can anyone explain choice A? thanks!

x > 2 is NOT necessarily true. Consider x = -1/2.
_________________
Intern
Joined: 24 Jun 2014
Posts: 49
Concentration: Social Entrepreneurship, Nonprofit
Re: If x/|x|, which of the following must be true for all  [#permalink]

### Show Tags

22 Apr 2015, 18:22
Brunel, Can you please explain why option E is not feasible?
Director
Joined: 07 Aug 2011
Posts: 538
Concentration: International Business, Technology
GMAT 1: 630 Q49 V27
Re: If x/|x|, which of the following must be true for all  [#permalink]

### Show Tags

22 Apr 2015, 19:19
1
praveenvino wrote:
If $$\frac{x}{|x|} \lt x$$, which of the following must be true about $$x$$? ($$x \ne 0$$)

A. $$x\gt 2$$
B. $$x \in (-1,0) \cup (1,\infty)$$
C. $$|x| \lt 1$$
D. $$|x| = 1$$
E. $$|x|^2 \gt 1$$

M24

x < x*|x|
x-x*|x|< 0
roots of this equation are : -1,0,1
rest is explained in the attached image ...
Answer B.
Attachments

gmatclub.jpg [ 16.88 KiB | Viewed 5249 times ]

_________________

Thanks,
Lucky

_______________________________________________________
Kindly press the to appreciate my post !!

e-GMAT Representative
Joined: 04 Jan 2015
Posts: 2291
If x/|x|, which of the following must be true for all  [#permalink]

### Show Tags

23 Apr 2015, 00:44
1
AverageGuy123 wrote:
Brunel, Can you please explain why option E is not feasible?

Dear AverageuGuy123

As Bunuel explained above,

Either -1 < x < 0
Or x > 1

Now, |x| as you know, represents the magnitude of x. Option E says that |x|^2 must be greater than 1.

Let's first consider the case when -1 < x < 0

A possible value of x in this case is -0.5
So, what is the value of |x|^2? It is equal to 0.25

Is it greater than 1? NO

Let's now consider the case when x > 1

A possible value of x in this case is 2.
So, what is the value of |x|^2? It's 4.

Is it greater than 1? YES

So, as we see, that |x|^2 CAN BE greater than 1. But can we say that |x|^2 MUST BE greater than 1? NO, because |x|^2 is not greater than 1 for all possible values of x.

So, the key takeaway from this discussion is that:

we need to be careful whether the question is asking about MUST BE TRUE statements or about CAN BE TRUE statements.

Hope this helped!

- Japinder
_________________

Register for free sessions
Number Properties | Algebra |Quant Workshop

Success Stories
Guillermo's Success Story | Carrie's Success Story

Ace GMAT quant
Articles and Question to reach Q51 | Question of the week

Must Read Articles
Number Properties – Even Odd | LCM GCD | Statistics-1 | Statistics-2 | Remainders-1 | Remainders-2
Word Problems – Percentage 1 | Percentage 2 | Time and Work 1 | Time and Work 2 | Time, Speed and Distance 1 | Time, Speed and Distance 2
Advanced Topics- Permutation and Combination 1 | Permutation and Combination 2 | Permutation and Combination 3 | Probability
Geometry- Triangles 1 | Triangles 2 | Triangles 3 | Common Mistakes in Geometry
Algebra- Wavy line | Inequalities

Practice Questions
Number Properties 1 | Number Properties 2 | Algebra 1 | Geometry | Prime Numbers | Absolute value equations | Sets

| '4 out of Top 5' Instructors on gmatclub | 70 point improvement guarantee | www.e-gmat.com

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8665
Location: Pune, India
Re: If x/|x|, which of the following must be true for all  [#permalink]

### Show Tags

23 Apr 2015, 03:36
AverageGuy123 wrote:
Brunel, Can you please explain why option E is not feasible?

You can plug in numbers to eliminate options.

"which of the following must be true about x" means that every acceptable value of x must lie in the range given in the correct option. The acceptable values of x are the values for which x/|x| < x.

A. x>2
Must x be greater than 2?

This should make you check for 2.
2/|2| < 2
1 < 2 (True)
So 2 is an acceptable value of x. But 2 is not greater than 2.
So this option is not correct. This also makes you eliminate options (C) and (D).

E. |x|^2>1
Must x be greater than 1 or less than -1?

Check for 1/2
(1/2)/|1/2| < 1/2
1 < 1/2 (False)

Check for -1/2
(-1/2)/|-1/2| < -1/2
-1 < -1/2 (True)

So x = -1/2 is an acceptable value but it does not lie in this range. Hence option (E) is also incorrect.

Answer must be (B)
_________________

Karishma
Veritas Prep GMAT Instructor

Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >

EMPOWERgmat Instructor
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 13064
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Re: If x/|x|, which of the following must be true for all  [#permalink]

### Show Tags

24 Apr 2015, 10:21
Hi All,

This question can be dealt with in a variety of ways. It's actually really susceptible to TESTing VALUES, which we can use to determine possibilities and eliminate answers.

We're told that X/|X| < X. The question asks what must be TRUE about X.

While this inequality looks complicated, you can quickly prove some things about X....

IF....
X = 1
1/|1| is NOT < 1
So X CANNOT be 1
Eliminate D.

IF.....
X = 2
2/|2| IS < 2
So X CAN be 2
Eliminate A and C.

IF....
X = -2
-2/|-2| is NOT < -2
So X CANNOT be -2
Eliminate E.

There's only one answer left....

Final Answer:

GMAT assassins aren't born, they're made,
Rich
_________________

760+: Learn What GMAT Assassins Do to Score at the Highest Levels
Contact Rich at: Rich.C@empowergmat.com

# Rich Cohen

Co-Founder & GMAT Assassin

Special Offer: Save \$75 + GMAT Club Tests Free
Official GMAT Exam Packs + 70 Pt. Improvement Guarantee
www.empowergmat.com/

*****Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!*****

Senior Manager
Joined: 02 Apr 2014
Posts: 474
GMAT 1: 700 Q50 V34
Re: If x/|x|, which of the following must be true for all  [#permalink]

### Show Tags

26 Jan 2018, 23:48
Given: x/|x| < x
since |x| >= 0 always
multiply both LHS and RHS by |x|
x < x|x|
=> x - x|x| < 0
=> x(1 - |x|) < 0
if x > 0, then 1 - |x| < 0 to hold the above inequality => |x| > 1 => x(since x is positive in this case) > 1
if x < 0, then 1 - |x| > 0 => |x| < 1 => -1 < x < 0 (to hold the above inequality)

Option B captures the above range perfectly
Re: If x/|x|, which of the following must be true for all &nbs [#permalink] 26 Jan 2018, 23:48
Display posts from previous: Sort by

# If x/|x|, which of the following must be true for all

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

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne 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®.