x/|x|<x. which of the following must be true about x ? : GMAT Problem Solving (PS) - Page 3
Check GMAT Club Decision Tracker for the Latest School Decision Releases http://gmatclub.com/AppTrack

 It is currently 22 Jan 2017, 07:08

### 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

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

Author Message
TAGS:

### Hide Tags

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

Kudos [?]: 24 [0], given: 1

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

### Show Tags

02 Jun 2014, 20:11
1
This post was
BOOKMARKED
Buneul, you are saying that x>-1, suppose we substitute 0 in the inequality we get 0<0. How is this true????
Math Expert
Joined: 02 Sep 2009
Posts: 36597
Followers: 7093

Kudos [?]: 93438 [0], given: 10563

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

### Show Tags

03 Jun 2014, 00:18
Buneul, you are saying that x>-1, suppose we substitute 0 in the inequality we get 0<0. How is this true????

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.
_________________
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 13500
Followers: 577

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

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

### Show Tags

02 Aug 2015, 10:06
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.
_________________
Intern
Joined: 03 Jul 2015
Posts: 40
Followers: 0

Kudos [?]: 11 [0], given: 27

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

### Show Tags

31 Aug 2015, 09:24
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.

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

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

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
Joined: 03 Jul 2015
Posts: 40
Followers: 0

Kudos [?]: 11 [0], given: 27

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

### Show Tags

31 Aug 2015, 09:38
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.

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

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

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

Kudos [?]: 11 [1] , given: 27

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

### Show Tags

31 Aug 2015, 10:14
1
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
Joined: 29 May 2015
Posts: 15
Followers: 0

Kudos [?]: 2 [0], given: 29

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

### Show Tags

15 Sep 2015, 00:18
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
Joined: 31 Oct 2015
Posts: 37
Followers: 0

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

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

### Show Tags

27 Dec 2015, 11:48
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)
BSchool Forum Moderator
Joined: 12 Aug 2015
Posts: 1902
Followers: 49

Kudos [?]: 369 [0], given: 455

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

### Show Tags

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

Mock Test -1 (Integer Properties Basic Quiz) ---> http://gmatclub.com/forum/stonecold-s-mock-test-217160.html#p1676182

Mock Test -2 (Integer Properties Advanced Quiz) --->http://gmatclub.com/forum/stonecold-s-mock-test-217160.html#p1765951

Mock Test -2 (Evensand Odds Basic Quiz) --->http://gmatclub.com/forum/stonecold-s-mock-test-217160.html#p1768023

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

Manager
Joined: 13 Apr 2016
Posts: 60
Location: India
GMAT 1: 640 Q50 V27
GPA: 3
WE: Operations (Hospitality and Tourism)
Followers: 0

Kudos [?]: 19 [0], given: 7

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

### Show Tags

19 May 2016, 04:20
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.

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

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

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.
Math Expert
Joined: 02 Sep 2009
Posts: 36597
Followers: 7093

Kudos [?]: 93438 [0], given: 10563

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

### Show Tags

19 May 2016, 06:32
Himanshu9818 wrote:
if answer to this question is B then why not x=1 satisfy the situation.

This is explained several times on previous pages. please go through the solutions provided before.
_________________
Intern
Joined: 13 Sep 2015
Posts: 29
Location: India
Schools: IIMA (I)
GMAT 1: 700 Q50 V34
GPA: 3.2
Followers: 0

Kudos [?]: 1 [0], given: 28

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

### Show Tags

26 May 2016, 19:10
X>-1 includes the case x=0 ...so x>1
Intern
Joined: 22 Sep 2016
Posts: 1
Followers: 0

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

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

### Show Tags

22 Sep 2016, 00:15
Hi friends! Saw that mainly there is a dispute between answer as A) or B).

As per me, answer is A). Here is the reasoning why B) cannot be an answer ( x > -1).

Take x=0.5 for an example. Then, You'll get the equation not being satisfied... So, x> -1 isn't a possibility... Hence, A is the answer.
Thanks

Posted from my mobile device
Math Expert
Joined: 02 Sep 2009
Posts: 36597
Followers: 7093

Kudos [?]: 93438 [0], given: 10563

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

### Show Tags

22 Sep 2016, 01:20
Uttunnu123 wrote:
Hi friends! Saw that mainly there is a dispute between answer as A) or B).

As per me, answer is A). Here is the reasoning why B) cannot be an answer ( x > -1).

Take x=0.5 for an example. Then, You'll get the equation not being satisfied... So, x> -1 isn't a possibility... Hence, A is the answer.
Thanks

Posted from my mobile device

There is no dispute. The correct answer is B. You can find several solutions on previous pages.
_________________
Manager
Joined: 24 Jul 2016
Posts: 97
Followers: 0

Kudos [?]: 4 [0], given: 5

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

### Show Tags

26 Sep 2016, 12:44
christoph 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

What if x=0? X=0>-1, but x > -1 must not be true.
_________________

Kaplan CAT#1 660 (Q50, V 33)
GMATPrep CAT#1 590 (Q 49, V 28)
Veritas CAT#1 640 (Q 47, V 31)
Veritas CAT#2 620 (Q 46, V 30)
Veritas CAT#3 680 (Q 50, V 34)
Veritas CAT#4 650 (Q 49, V 31)
Princeton Review CAT#1 650 (Q 44, V 36)
Kaplan CAT2 720 (Q49, V40)- got some questions in other tests already
Veritas CAT#5 680
Kaplan CAT#3 710
Target GMAT 720

Manager
Joined: 26 Jan 2016
Posts: 111
Location: United States
GMAT 1: 700 Q48 V39
GPA: 3.37
Followers: 1

Kudos [?]: 19 [0], given: 55

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

### Show Tags

26 Sep 2016, 13:42
I was down to B and C. Can someone explain why C is wrong? I can't find an example where B is right but C is wrong.
Manager
Joined: 24 Jul 2016
Posts: 97
Followers: 0

Kudos [?]: 4 [0], given: 5

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

### Show Tags

26 Sep 2016, 13:45
Take X=2, c will be wrong

Sent from my iPhone using GMAT Club Forum mobile app
_________________

Kaplan CAT#1 660 (Q50, V 33)
GMATPrep CAT#1 590 (Q 49, V 28)
Veritas CAT#1 640 (Q 47, V 31)
Veritas CAT#2 620 (Q 46, V 30)
Veritas CAT#3 680 (Q 50, V 34)
Veritas CAT#4 650 (Q 49, V 31)
Princeton Review CAT#1 650 (Q 44, V 36)
Kaplan CAT2 720 (Q49, V40)- got some questions in other tests already
Veritas CAT#5 680
Kaplan CAT#3 710
Target GMAT 720

Intern
Joined: 02 Jun 2013
Posts: 2
Followers: 0

Kudos [?]: 0 [0], given: 11

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

### Show Tags

10 Oct 2016, 04:58
Ans. B
Reason:
x/|x|<x => x/|x| - x <0 => x(1-|x|)/|x| <0

For+ve x,
1-|x| must be less than 0 i.e. 1-|x| <0 => x>1

For-ve x,
1-|x|>0 => -1<x <0

From this we can say that in both cases, -1<x
Intern
Joined: 08 May 2016
Posts: 10
Followers: 0

Kudos [?]: 1 [0], given: 4

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

### Show Tags

12 Oct 2016, 23:08
christoph 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

Bunuel, I did it this way, please tell me if there is anything wrong in my approach?

On the LHS we have: x/|x|. This value will either be +1 or -1 depending on the value of x.

Therefore, 1<x or -1<x. The larger range of the two is 1-<x, hence I chose Option B as it covers both ranges.
Intern
Joined: 24 Aug 2016
Posts: 21
Location: India
WE: Project Management (Aerospace and Defense)
Followers: 0

Kudos [?]: 6 [0], given: 27

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

### Show Tags

14 Oct 2016, 19:17
christoph 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

It looks the value |x| > 1 is the right answer, Thus A is the correct one
Re: x/|x|<x. which of the following must be true about x ?   [#permalink] 14 Oct 2016, 19:17

Go to page   Previous    1   2   3   4    Next  [ 62 posts ]

Similar topics Replies Last post
Similar
Topics:
3 Which of the following is true about 0<|x|-4x<5? 4 26 Apr 2016, 05:33
4 If √x=x , then which of the following must be true ? 3 26 Oct 2014, 03:35
18 If |x|=−x, which of the following must be true? 5 22 Mar 2014, 01:31
28 If x/|x|, which of the following must be true for all 14 15 Jan 2011, 11:44
2 If x/|x| < x, which of the following must be true about 25 06 Sep 2009, 21:14
Display posts from previous: Sort by