Check GMAT Club Decision Tracker for the Latest School Decision Releases https://gmatclub.com/AppTrack

 It is currently 29 May 2017, 18:32

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

Author Message
TAGS:

### Hide Tags

Senior Manager
Joined: 06 Apr 2008
Posts: 437
Followers: 1

Kudos [?]: 155 [24] , given: 1

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

### Show Tags

15 Aug 2008, 04:06
24
KUDOS
93
This post was
BOOKMARKED
00:00

Difficulty:

95% (hard)

Question Stats:

33% (01:54) correct 67% (01:12) wrong based on 2449 sessions

### HideShow timer Statistics

If $$\frac{x}{|x|}<x$$ which of the following must be true about $$x$$?

(A) $$x>1$$

(B) $$x>-1$$

(C) $$|x|<1$$

(D) $$|x|=1$$

(E) $$|x|^2>1$$
[Reveal] Spoiler: OA
Math Expert
Joined: 02 Sep 2009
Posts: 39062
Followers: 7756

Kudos [?]: 106577 [39] , given: 11628

### Show Tags

18 Jan 2010, 04:11
39
KUDOS
Expert's post
20
This post was
BOOKMARKED
nmohindru wrote:
If $$\frac{x}{|x|}<x$$ which of the following must be true about $$x$$?

(A) $$x>1$$

(B) $$x>-1$$

(C) $$|x|<1$$

(D) $$|x|=1$$

(E) $$|x|^2>1$$

This question was well explained by Durgesh and Ian Stewart, but since there are still some doubts, I'll try to add my 2 cents.

First of all let's solve this inequality step by step and see what is the solution for it, or in other words let's see in which ranges this inequality holds true.

Two cases for $$\frac{x}{|x|}<x$$:

A. $$x<0$$ --> $$|x|=-x$$ --> $$\frac{x}{-x}<x$$ --> $$-1<x$$ --> $$-1<x<0$$;

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

So given inequality holds true in the ranges: $$-1<x<0$$ and $$x>1$$. Which means that $$x$$ can take values only from these ranges.

------{-1}xxxx{0}----{1}xxxxxx

Now, we are asked which of the following must be true about $$x$$. Option A can not be ALWAYS true because $$x$$ can be from the range $$-1<x<0$$, eg $$-\frac{1}{2}$$ and $$x=-\frac{1}{2}<1$$.

Only option which is ALWAYS true is B. ANY $$x$$ from the ranges $$-1<x<0$$ and $$x>1$$ will definitely be more the $$-1$$, all "red", possible x-es are to the right of -1, which means that all possible x-es are more than -1.

_________________
GMAT Tutor
Joined: 24 Jun 2008
Posts: 1179
Followers: 438

Kudos [?]: 1606 [17] , given: 4

### Show Tags

17 Aug 2008, 08:29
17
KUDOS
Expert's post
7
This post was
BOOKMARKED
I'm pretty sure Durgesh has already explained his answer convincingly, but just in case, I'll present a different problem:

If x is positive, what must be true?

I) x > 10
II) x > -10
III) x > 0

Of course, III) says exactly 'x is positive', so III) must be true. But if x is positive, x is obviously larger than -10. II) must also be true. I) doesn't need to be true; x could be 4, or 7, for example.

The same situation applies with the question that began this thread. We know that either -1 < x < 0, or x > 1. What must be true? Well, "x > 1" does not need to be true. We know that x might be -0.5, for example. On the other hand, "x > -1" absolutely must be true: if x satisfies the inequality x/|x| < x, then x is certainly larger than -1. If we had been asked whether "x > -1,000,000" must be true, the answer would also be yes. The question did not ask "What is the solution set of "x/|x| < x"; nor did it ask "For which values of x is x/|x| < x true?".
_________________

GMAT Tutor in Toronto

If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com

SVP
Joined: 07 Nov 2007
Posts: 1805
Location: New York
Followers: 38

Kudos [?]: 934 [15] , given: 5

### Show Tags

16 Aug 2008, 14:22
15
KUDOS
For x < 0,

x / |x| < x
=> x / -x < x
=> x > -1
x < 0 & x > -1
-1<X<0

For x > 0,
x / |x| < x
=> x / x < x
=> x > 1
x>0&x > 1
----> x>1

Question says which one must be true..

Option A ) correct.. X>1 (X>1 implies that when X>0 )
this is always true

bet on A.
_________________

Smiling wins more friends than frowning

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7380
Location: Pune, India
Followers: 2292

Kudos [?]: 15168 [12] , given: 224

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

### Show Tags

13 Jan 2012, 10:46
12
KUDOS
Expert's post
gautammalik wrote:
I think answer should be A and not B because x cannot be equal to zero. If x equals to zero then the equation will lead to infinity.

Think again.
Every value greater than -1 need not satisfy the inequality but every value satisfying the inequality must be greater than -1.

x/|x| can take only 2 values: 1 or -1
If x is positive, x/|x| = 1
If x is negative, x/|x| = -1
x cannot be 0.

Now let's look at the question.
x > x/|x| holds for x > 1 (x is positive) or -1 < x < 0 (x is negative)

x can take many values e.g. -1/3, -4/5, 2, 5, 10 etc

Which of the following holds for every value that x can take?
(A) X > 1
(B) X > -1

I hope that you agree that X > 1 doesn't hold for every possible value of X whereas X > -1 holds for every possible value of X.

Mind you, every value greater than -1 need not be a possible value of x.
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Director Joined: 27 May 2008 Posts: 544 Followers: 8 Kudos [?]: 336 [11] , given: 0 Re: Inequality [#permalink] ### Show Tags 15 Aug 2008, 08:37 11 This post received KUDOS 1 This post was BOOKMARKED i think OA is correct.... the question is If x / |x| < x. so we have to consider only those values of x for which this inequalty is true and what are those values 1. when x is between -1 and 0 2. when x is more than 1 lets call these conditions our universe. Now the question is for all values of x (in our universe) which of the following is true option B, x > -1, has both conditions 1 and 2 now you may say what about x = 1/2 .... that wasnt even part of our universe... so even if x = 1/2 is satisfying option B and not the question stem, we dont have to worry... becuase we are not supposed to take it as an example ... for all values of x in our universe, option B is ALWAYS true... Option A is not always true... SVP Joined: 30 Apr 2008 Posts: 1874 Location: Oklahoma City Schools: Hard Knocks Followers: 42 Kudos [?]: 590 [7] , given: 32 Re: Inequality [#permalink] ### Show Tags 15 Aug 2008, 07:27 7 This post received KUDOS To condense some of what the other thread states: If you use -1/2, it works, so at first you might think the answer is clearly b), but "must be true about X" also means "must ALWAYS be true about X". If you use +1/2 (which is > -1 as in answer B) it doesn't work. 1/2 over 1/2 = 1 < x which is 1/2 so you get 1 < 1/2 and that's not true. So answer a MUST ALWAYS be true because only positive X over abs(x) = 1 and X > 1, it will always be greater than 1 (which is x/|x| when x is positive). Does this condense the other thread? nmohindru wrote: If 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 _________________ ------------------------------------ J Allen Morris **I'm pretty sure I'm right, but then again, I'm just a guy with his head up his a$$. GMAT Club Premium Membership - big benefits and savings SVP Joined: 30 Apr 2008 Posts: 1874 Location: Oklahoma City Schools: Hard Knocks Followers: 42 Kudos [?]: 590 [2] , given: 32 Re: Inequality [#permalink] ### Show Tags 16 Aug 2008, 13:56 2 This post received KUDOS Because the question states what MUST be true, I think the answer has to be A. We can't select only our "universe" of numbers that satisfy the selection. If the answer, like in B, says X > -1, then every single number greater than -1 must satisfy the inequality and that is not the case. I certainly agree that the "universe" must be -1 < X < 0 and 1 < x. Answer B defines one "universe" as all numbers that are greater than -1. Answer B includes all numbers between 0 and 1 which do not satisfy the inequality; therefore, B must not always be true. The answer should be A. I disagree with the OA. nmohindru wrote: If X/|X| < X Which of the following must be true about X ? X>1 X>-1 |X| < 1 |X| = 1 |X|^2 > 1 _________________ ------------------------------------ J Allen Morris **I'm pretty sure I'm right, but then again, I'm just a guy with his head up his a$$. GMAT Club Premium Membership - big benefits and savings Director Joined: 27 May 2008 Posts: 544 Followers: 8 Kudos [?]: 336 [2] , given: 0 Re: Inequality [#permalink] ### Show Tags 16 Aug 2008, 17:21 2 This post received KUDOS 2 This post was BOOKMARKED guys i beg to differ.. i'll try to explain myself again... our universe is defined in this line, if x/|x| < x this is given ... x only exits if it follows this condition ... if soemthing doesnt follow it then, its not x ... its y, may be z (doesnt matter) ... if S is a set of possible values of x that will satisfy our condition x/|x| < x S = [ -0.9, -0.8, -0.7, ....... -0.1, 1.1, 1.2, 1.3, 1.4 ....... upto infinite] we have to find an answer that will always be true for all values of x, lets see A, x > 1, this statement is not true for x = -0.9, -0.8 ..... -0.1 check B x > -1, this statement will "always" be true for all x ... clearly B is the answer... i see your point that x = 0.5 will not satisfy the question condition, but thats the whole point 0.5 is not even x, its y but if you select A, you are leaving some valid values of x. Director Joined: 27 May 2008 Posts: 544 Followers: 8 Kudos [?]: 336 [2] , given: 0 Re: Inequality [#permalink] ### Show Tags 16 Aug 2008, 18:12 2 This post received KUDOS GMAT TIGER wrote: x = 0, 1 , or any +ve fractions. that is not even defined as x, our x is defined in the first line .... i'll give an example, x is a member of set S = [-4,-3,-2,-1, 4,5,6,7] what is always true for all x A. x >= 4 B. x >= -4 A or B... if you select A, then i cant explain it further.. GMAT Tutor Joined: 24 Jun 2008 Posts: 1179 Followers: 438 Kudos [?]: 1606 [2] , given: 4 Re: If x/|x|<x which of the following must be true about x? [#permalink] ### Show Tags 20 Dec 2011, 03:08 2 This post received KUDOS Expert's post 2 This post was BOOKMARKED myfish wrote: Typical GMAT nonsense. A lot of people seem to talk themselves into a solution but in mathematics, there are no GMAT-truths. x>-1 must not be true, since it ignores the fact that 0 does not fulfill the requirement. No one is "talking themselves into a solution" here, and there's nothing wrong with the mathematics. I explained why earlier, but I can use a simpler example. If a question reads If x = 5, what must be true? I) x > 0 then clearly I) must be true; if x is 5, then x is certainly positive. It makes no difference that x cannot be equal to 12, or to 1000. The same thing is happening in this question. We know that either -1 < x < 0, or that 1 < x. If x is in either of those ranges, then certainly x must be greater than -1. It makes no difference that x cannot be equal to 1/2, or to 0. This is an important logical point on the GMAT (even though the question in the original post is not a real GMAT question), since it comes up all the time in Data Sufficiency. If a question asks Is x > 0? 1) x = 5 that is exactly the same question as the one I asked above, but now it's phrased as a DS question. This question is really asking, when we use Statement 1, "If x = 5, must it be true that x > 0?" Clearly the answer is yes. If you misinterpret this question, and think it's asking "can x have any positive value at all", you would make a mistake on this question and on most GMAT DS algebra questions. _________________ GMAT Tutor in Toronto If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7380 Location: Pune, India Followers: 2292 Kudos [?]: 15168 [2] , given: 224 Re: If x/|x|<x which of the following must be true about x? [#permalink] ### Show Tags 21 Feb 2016, 22:14 2 This post received KUDOS Expert's post 1 This post was BOOKMARKED happyface101 wrote: Engr2012 wrote: happyface101 wrote: Hi Karishma and other experts - I understand the solution but I'm still stuck on why reasoning is wrong. This is how I solved: x/|x| < x --> x/|x| -x --> x*(1/|x|-1)<0, so: 1. x<0; (1/|x|-1)<0 -> 1/|x|<1 -> 1<|x| -> x<-1 THIS IS WHERE IT'S WRONG BUT WHY? 2. x>0; (1/|x|-1)>0 -> 1/|x|>1 -> 1>|x| -> 1>x THIS IS WHERE IT'S WRONG BUT WHY? Thank you! The mistake you are making is in the portion with red text above. Your 'simplified' expression is x(1/|x|-1) and not just (1/|x|-1). Thus your cases become: Case 1: x<0 --> |x|=-x ---> x(1/|x|-1)<0 ---> x(-1/x-1)<0 ---->x(1/x+1)>0 --->x(1+x)/x > 0--->x+1>0 --->x>-1 and this with the fact that x<0 ---> range becomes -1<x<0. Case 2: x>0 --> |x|=x ---> x(1/|x|-1)<0 ---> x(1/x-1)<0 ---->x(1/x-1)<0 --->x(1-x)/x < 0--->1-x<0 --->x>1 and this with the fact that x>0 ---> range becomes x>1. Hope this helps. Thank you so much! +1 I think the knowledge gap is that I treated x*(1/|x|-1)<0 as if it's x*((1/|x|-1)=0 In an inequality problem I obviously can't solve by taking the two apart and do x<0, (1/|x|-1)<0 like I can for an equation. Thanks for helping me understand this! Additionally, you can split the factors as you did but you made an error there. x*(1/|x|-1)<0 implies that x*(1/|x|-1) is negative. So either x is negative and (1/|x|-1) is positive or x is positive and (1/|x|-1) is negative Case 1: x is negative and (1/|x|-1) is positive When x is negative, |x| = -x (1/|x|-1) > 0 (1/(-x) - 1) > 0 1/x + 1 < 0 (1+x)/x < 0 -1 < x< 0 Case 2: x is positive and (1/|x|-1) is negative When x is positive, |x| = x (1/|x|-1) < 0 (1/x - 1) < 0 (1 - x)/x < 0 (x - 1)/x > 0 x > 1 or x < 0 But x is positive so x cannot be less than 0. So x > 1. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

Veritas Prep Reviews

Intern
Joined: 21 Jul 2008
Posts: 5
Followers: 0

Kudos [?]: 2 [1] , given: 0

### Show Tags

16 Aug 2008, 12:50
1
KUDOS
1
This post was
BOOKMARKED
I believe (E) should be the answer.

we know that |x| = -x for x < 0
= x for x > 0

For x < 0,

x / |x| < x
=> x / -x < x
=> x > -1

For x > 0,

x / |x| < x
=> x / x < x
=> x > 1

Therefore |x| > 1 or |x|^2 > 1
Director
Joined: 27 May 2008
Posts: 544
Followers: 8

Kudos [?]: 336 [1] , given: 0

### Show Tags

16 Aug 2008, 17:50
1
KUDOS
GMAT TIGER wrote:
nmohindru wrote:
If X/|X| < X Which of the following must be true about X ?

X>1

X>-1

|X| < 1

|X| = 1

|X|^2 > 1

For me it is disputable. I sometime go with B and sometime with A. B is broader than A but B again is not always a true case where as A is always true.

if we pay detail attention to MUST on "Which if the following must be true about X", it should be A because B is not a "must" true.

Please go through my earlier posts and give me a value of x for which B is not true, i dont think you can find one ....
Manager
Joined: 29 Jul 2011
Posts: 107
Location: United States
Followers: 5

Kudos [?]: 64 [1] , given: 6

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

### Show Tags

13 Jan 2012, 12:17
1
KUDOS
What this means is that x could be negative fraction or positive integer/fraction. Try plugging in -2, -1, -0.8, 1, 2, 3 ....

B
_________________

I am the master of my fate. I am the captain of my soul.
Please consider giving +1 Kudos if deserved!

DS - If negative answer only, still sufficient. No need to find exact solution.
PS - Always look at the answers first
CR - Read the question stem first, hunt for conclusion
SC - Meaning first, Grammar second
RC - Mentally connect paragraphs as you proceed. Short = 2min, Long = 3-4 min

Math Expert
Joined: 02 Sep 2009
Posts: 39062
Followers: 7756

Kudos [?]: 106577 [1] , given: 11628

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

### Show Tags

27 May 2012, 05:16
1
KUDOS
Expert's post
vibhav wrote:
Bunuel, the question doesn't mention anywhere that x is an integer. so why have we not considered the values of 0<x<1 which also doesnt satisfy the equation? If it has been considered then the solution should be A and not B. x>1 will always hold true aka must be true! while x>-1 is sometimes true.. aka can be true.

I think you don't understand the question.

Given: $$-1<x<0$$ and $$x>1$$. Question: which of the following must be true?

A. $$x>1$$. This opinion is not always true since $$x$$ can be $$-\frac{1}{2}$$ which is not more than 1.
B. $$x>-1$$. This option is always true since any $$x$$ from $$-1<x<0$$ and $$x>1$$ is more than -1.
_________________
Manager
Joined: 08 Apr 2012
Posts: 129
Followers: 12

Kudos [?]: 103 [1] , given: 14

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

### Show Tags

30 May 2012, 05:36
1
KUDOS
pavanpuneet wrote:
Hi Shouvik,

But I am not dividing .. I first consider that x<0 then cross multiply and flip the sign. I am still not clear with where did I go wrong.

Ok,

Let me explain step by step:

Note that we have considered x<0. So anything we divide or multiply by a negative number x will change the signs of an inequality.

1. Since x<0, x/|x| < x => x/(-x) < x
2. Cross-multiplying both sides by (-x). Now since x<0, (-x)>0. So if we cross multiply it doesn't change the sign.
x < -x^2
3. Now, we divide both sides by x. Since x<0, this changes the sign of the inequality.
1 > -x
4. Simplifying further,
-1<x

This is exactly what bunuel had got.

Regards,

Shouvik.
_________________

Shouvik
http://www.Edvento.com

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7380
Location: Pune, India
Followers: 2292

Kudos [?]: 15168 [1] , given: 224

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

### Show Tags

12 Jun 2012, 04:31
1
KUDOS
Expert's post
There was a lot of confusion between options (A) and (B). Therefore, I would like to explain why option (B) is correct using diagrams.

Forget this question for a minute. Say instead you have this question:

Question 1: x > 2 and x < 7. What integral values can x take?
I guess most of you will come up with 3, 4, 5, 6. That’s correct. I can represent this on the number line.
Attachment:

Ques3.jpg [ 4.45 KiB | Viewed 7762 times ]

You see that the overlapping area includes 3, 4, 5 and 6.

Now consider this:

Question 2: x > 2 or x > 5. What integral values can x take?
Let’s draw that number line again.
Attachment:

Ques4.jpg [ 4.24 KiB | Viewed 7758 times ]

So is the solution again the overlapping numbers i.e. all integers greater than 5? No. This question is different. x is greater than 2 OR greater than 5. This means that if x satisfies at least one of these conditions, it is included in your answer. Think of sets. AND means it should be in both the sets (i.e. overlapping). OR means it should be in at least one of the sets. Hence, which values can x take? All integral values starting from 3 onwards i.e. 3, 4, 5, 6, 7, 8, 9 …

Now go back to this question. The solution is a one liner.

If $$\frac{x}{|x|}<x$$ which of the following must be true about $$x$$?
(A) $$x>1$$
(B) $$x>-1$$
(C) $$|x|<1$$
(D) $$|x|=1$$
(E) $$|x|^2>1$$

$$\frac{x}{|x|}$$ is either 1 or -1.
So x > 1 or x > -1
So which values can x take? All values that are included in at least one of the sets. Therefore, x > -1.
_________________

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: 39062
Followers: 7756

Kudos [?]: 106577 [1] , given: 11628

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

### Show Tags

15 Aug 2014, 09:28
1
KUDOS
Expert's post
sri30kanth wrote:
Bunuel,

What if we square the inequality x/ |x| < x. Then we get (x^2 / x ) < x^2 which implies that x^2 < x^3. Is this correct? Please explain. Thanku

We can raise both parts of an inequality to an even power if we know that both parts of an inequality are non-negative (the same for taking an even root of both sides of an inequality), which is not the case here.

Also, the second step in your solution: never multiply (or reduce) an inequality by a variable (or the expression with a variable) if you don't know the sign of it, we don't know the sign of x, so we cannot multiply x^2/x < x^2 by x here.

For more check here: inequalities-tips-and-hints-175001.html
_________________
Math Expert
Joined: 02 Sep 2009
Posts: 39062
Followers: 7756

Kudos [?]: 106577 [1] , given: 11628

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

### Show Tags

17 Aug 2014, 03:58
1
KUDOS
Expert's post
dransa wrote:
nmohindru wrote:
If $$\frac{x}{|x|}<x$$ which of the following must be true about $$x$$?

(A) $$x>1$$

(B) $$x>-1$$

(C) $$|x|<1$$

(D) $$|x|=1$$

(E) $$|x|^2>1$$

I disagree with OA .According to me the Answer should be A.
OA is wrong for insistence take x=1/2 (since 1/2 > -1) but the inequality does not hold true for this value, so how can the answer be correct.

_________________
Re: If x/|x|<x which of the following must be true about x?   [#permalink] 17 Aug 2014, 03:58

Go to page    1   2   3   4   5    Next  [ 98 posts ]

Similar topics Replies Last post
Similar
Topics:
3 Which of the following is true about 0<|x|-4x<5? 4 02 May 2016, 02:59
5 If √x=x , then which of the following must be true ? 3 29 Mar 2016, 06:20
20 If |x|=−x, which of the following must be true? 7 04 Apr 2017, 07:29
29 If x/|x|, which of the following must be true for all 14 30 Jun 2016, 10:48
2 If x/|x| < x, which of the following must be true about 25 26 Feb 2015, 00:41
Display posts from previous: Sort by