It is currently 18 Oct 2017, 02:43

### 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

# M09-22

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 41886

Kudos [?]: 128668 [0], given: 12181

### Show Tags

16 Sep 2014, 00:40
Expert's post
22
This post was
BOOKMARKED
00:00

Difficulty:

95% (hard)

Question Stats:

28% (01:04) correct 72% (01:11) wrong based on 204 sessions

### HideShow timer Statistics

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

A. $$x \gt 1$$
B. $$x \gt -1$$
C. $$|x| \lt 1$$
D. $$|x|&gt;1$$
E. $$-1 \lt x \lt 0$$
[Reveal] Spoiler: OA

_________________

Kudos [?]: 128668 [0], given: 12181

Math Expert
Joined: 02 Sep 2009
Posts: 41886

Kudos [?]: 128668 [0], given: 12181

### Show Tags

16 Sep 2014, 00:40
Expert's post
7
This post was
BOOKMARKED
Official Solution:

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

A. $$x \gt 1$$
B. $$x \gt -1$$
C. $$|x| \lt 1$$
D. $$|x|&gt;1$$
E. $$-1 \lt x \lt 0$$

Notice that we are asked to find which of the options MUST be true, not COULD be true.

Let's see what ranges does $$\frac{x}{|x|} \lt x$$ give for $$x$$. Two cases:

If $$x \lt 0$$ then $$|x|=-x$$, hence in this case we would have: $$\frac{x}{-x} \lt x$$, which is the same as $$-1 \lt x$$. But remember that we consider the range $$x \lt 0$$, so $$-1 \lt x \lt 0$$;

If $$x \gt 0$$ then $$|x|=x$$, hence in this case we would have: $$\frac{x}{x} \lt x$$, which is the same as $$1 \lt x$$.

So, $$\frac{x}{|x|} \lt x$$ means that $$-1 \lt x \lt 0$$ or $$x \gt 1$$.

Only option which is ALWAYS true is B. ANY $$x$$ from the range $$-1 \lt x \lt 0$$ or $$x \gt 1$$ will definitely be more the $$-1$$.

As for other options:

A. $$x \gt 1$$. Not necessarily true since $$x$$ could be -0.5;

C. $$|x| \lt 1$$, so $$-1 \lt x \lt 1$$. Not necessarily true since $$x$$ could be 2;

D. $$|x| \gt 1$$, so $$x \lt -1$$ or $$x \gt 1$$. Not necessarily true since $$x$$ could be -0.5;

E. $$-1 \lt x \lt 0$$. Not necessarily true since $$x$$ could be 2.

_________________

Kudos [?]: 128668 [0], given: 12181

Manager
Joined: 27 May 2014
Posts: 94

Kudos [?]: 105 [1], given: 43

Location: India
Concentration: Technology, General Management
Schools: HKUST '15, ISB '15
GMAT Date: 12-26-2014
GPA: 3

### Show Tags

15 Oct 2014, 00:20
1
KUDOS
Hi Bunuel

If i put X= 0.5 which is X > -1, Then the
L.H.S = 1
And R.H.S = 0.5.
R.H.S is not greater than L.H.S
_________________

Success has been and continues to be defined as Getting up one more time than you have been knocked down.

Kudos [?]: 105 [1], given: 43

Math Expert
Joined: 02 Sep 2009
Posts: 41886

Kudos [?]: 128668 [1], given: 12181

### Show Tags

15 Oct 2014, 00:24
1
KUDOS
Expert's post
1
This post was
BOOKMARKED
VarunBhardwaj wrote:
Hi Bunuel

If i put X= 0.5 which is X > -1, Then the
L.H.S = 1
And R.H.S = 0.5.
R.H.S is not greater than L.H.S

x cannot be 0.5 because it does not satisfy given condition that $$\frac{x}{|x|} \lt x$$. Please re-read the solution.
_________________

Kudos [?]: 128668 [1], given: 12181

Manager
Joined: 27 May 2014
Posts: 94

Kudos [?]: 105 [0], given: 43

Location: India
Concentration: Technology, General Management
Schools: HKUST '15, ISB '15
GMAT Date: 12-26-2014
GPA: 3

### Show Tags

15 Oct 2014, 03:40
Apologies.
Thanks for explaining. I completely misunderstood the question stem.
_________________

Success has been and continues to be defined as Getting up one more time than you have been knocked down.

Kudos [?]: 105 [0], given: 43

Intern
Joined: 24 Jun 2015
Posts: 46

Kudos [?]: 2 [0], given: 22

### Show Tags

08 Jul 2015, 08:13
Bunuel wrote:
VarunBhardwaj wrote:
Hi Bunuel

If i put X= 0.5 which is X > -1, Then the
L.H.S = 1
And R.H.S = 0.5.
R.H.S is not greater than L.H.S

x cannot be 0.5 because it does not satisfy given condition that $$\frac{x}{|x|} \lt x$$. Please re-read the solution.

Hi Bunuel,

I am confused with the MUST be true condition, I understand that MUST be true is like SUFFICIENCY for DS problems, I do not know If I am wrong or right but for instance... If X > -1 then in a DS question the answer is YES for -1 < x < 0 and x > 1, but the answer is NO for 0 < x < 1... then it would not be sufficient... I understand that MUST is like in ALL cases the same answer (In all cases affirm the question stem), but I am confused know, could you help me?

Thanks a lot.

Regards.

Luis Navarro
Looking for 700

Kudos [?]: 2 [0], given: 22

Math Expert
Joined: 02 Sep 2009
Posts: 41886

Kudos [?]: 128668 [0], given: 12181

### Show Tags

08 Jul 2015, 08:48
luisnavarro wrote:
Bunuel wrote:
VarunBhardwaj wrote:
Hi Bunuel

If i put X= 0.5 which is X > -1, Then the
L.H.S = 1
And R.H.S = 0.5.
R.H.S is not greater than L.H.S

x cannot be 0.5 because it does not satisfy given condition that $$\frac{x}{|x|} \lt x$$. Please re-read the solution.

Hi Bunuel,

I am confused with the MUST be true condition, I understand that MUST be true is like SUFFICIENCY for DS problems, I do not know If I am wrong or right but for instance... If X > -1 then in a DS question the answer is YES for -1 < x < 0 and x > 1, but the answer is NO for 0 < x < 1... then it would not be sufficient... I understand that MUST is like in ALL cases the same answer (In all cases affirm the question stem), but I am confused know, could you help me?

Thanks a lot.

Regards.

Luis Navarro
Looking for 700

Go through the discussion of the same question here: if-x-0-and-x-x-x-which-of-the-following-must-be-true-143572.html

Also, practice must or could be true questions here: search.php?search_id=tag&tag_id=193

Hope it helps.
_________________

Kudos [?]: 128668 [0], given: 12181

Intern
Joined: 23 Aug 2015
Posts: 2

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

### Show Tags

17 Nov 2015, 12:29
I think this is a high-quality question.

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

Manager
Joined: 08 Jul 2015
Posts: 60

Kudos [?]: 19 [0], given: 51

GPA: 3.8
WE: Project Management (Energy and Utilities)

### Show Tags

16 Jun 2016, 04:53
valmikee wrote:
I think this the explanation isn't clear enough, please elaborate. This given condition would not hold true if x=1. Hence, I don't think option B is the appropriate choice. Can someone please help clarify?

valmikee: the inequality gives us either $$x >1$$ or $$-1<x<0$$

Hence, in any case of satisfied solution of inequality, x will always be greater than -1 --> must be true that: $$x >-1$$ (B)
_________________

[4.33] In the end, what would you gain from everlasting remembrance? Absolutely nothing. So what is left worth living for?
This alone: justice in thought, goodness in action, speech that cannot deceive, and a disposition glad of whatever comes, welcoming it as necessary, as familiar, as flowing from the same source and fountain as yourself. (Marcus Aurelius)

Kudos [?]: 19 [0], given: 51

Intern
Joined: 13 Jun 2016
Posts: 19

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

### Show Tags

21 Jun 2016, 19:57
this question is amazing. im trying to figure out - how does answer E potentially result in the value 2 if x cannot take on two distinctive values? would you mind giving me a simple example with values? thank you

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

Manager
Joined: 08 Jul 2015
Posts: 60

Kudos [?]: 19 [0], given: 51

GPA: 3.8
WE: Project Management (Energy and Utilities)

### Show Tags

21 Jun 2016, 20:41
lydennis8 wrote:
this question is amazing. im trying to figure out - how does answer E potentially result in the value 2 if x cannot take on two distinctive values? would you mind giving me a simple example with values? thank you

lydennis8 - maybe some misunderstanding?

I think Bunuel only said that (E) not necessary true since if x = 2 then the inequality is still hold.

Say x= 3 then: $$3/|3|=1 < 3,$$ (correct) and so on...

Hope it's clear
_________________

[4.33] In the end, what would you gain from everlasting remembrance? Absolutely nothing. So what is left worth living for?
This alone: justice in thought, goodness in action, speech that cannot deceive, and a disposition glad of whatever comes, welcoming it as necessary, as familiar, as flowing from the same source and fountain as yourself. (Marcus Aurelius)

Kudos [?]: 19 [0], given: 51

Intern
Joined: 13 Jun 2016
Posts: 19

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

### Show Tags

21 Jun 2016, 23:35
Linhbiz thanks for that, misunderstood the wording

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

Intern
Joined: 15 Jul 2013
Posts: 1

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

### Show Tags

21 Jul 2016, 05:28
Hello, thanks for interesting question.

how can |x| = -x when x<0? if x <0 then |-x|=x?

I think when x<0, -x/|x|<-x. isn't it?

Please explain the flaws in my logic.

thank u!

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

Math Expert
Joined: 02 Sep 2009
Posts: 41886

Kudos [?]: 128668 [0], given: 12181

### Show Tags

21 Jul 2016, 08:24
Expert's post
2
This post was
BOOKMARKED
lshoshiashvili wrote:
Hello, thanks for interesting question.

how can |x| = -x when x<0? if x <0 then |-x|=x?

I think when x<0, -x/|x|<-x. isn't it?

Please explain the flaws in my logic.

thank u!

This is a property of an absolute value:

When $$x\leq{0}$$ then $$|x|=-x$$, or more generally when $$some \ expression\leq{0}$$ then $$|some \ expression|={-(some \ expression)}$$. For example: $$|-5|=5=-(-5)$$;

When $$x\geq{0}$$ then $$|x|=x$$, or more generally when $$some \ expression\geq{0}$$ then $$|some \ expression|={some \ expression}$$. For example: $$|5|=5$$.

You should brush-up fundamentals on modulus:

Theory on Absolute Values: math-absolute-value-modulus-86462.html
The E-GMAT Question Series on ABSOLUTE VALUE: the-e-gmat-question-series-on-absolute-value-198503.html
Properties of Absolute Values on the GMAT: properties-of-absolute-values-on-the-gmat-191317.html
Absolute Value: Tips and hints: absolute-value-tips-and-hints-175002.html

DS Absolute Values Questions to practice: search.php?search_id=tag&tag_id=37
PS Absolute Values Questions to practice: search.php?search_id=tag&tag_id=58

Hard set on Absolute Values: inequality-and-absolute-value-questions-from-my-collection-86939.html

_________________

Kudos [?]: 128668 [0], given: 12181

Senior Manager
Joined: 31 Mar 2016
Posts: 406

Kudos [?]: 78 [0], given: 197

Location: India
Concentration: Operations, Finance
GMAT 1: 670 Q48 V34
GPA: 3.8
WE: Operations (Commercial Banking)

### Show Tags

07 Aug 2016, 02:55
I think this is a high-quality question and I agree with explanation.

Kudos [?]: 78 [0], given: 197

Manager
Joined: 10 Feb 2017
Posts: 62

Kudos [?]: 17 [0], given: 64

Location: India
GMAT 1: 680 Q50 V30
GPA: 3.9

### Show Tags

20 Mar 2017, 23:15
1
This post was
BOOKMARKED
how can the answer is B
as clearly x=0.5 doesn't satisfies the equation.

i agree that x>1 or -1<x<0 but this doesn't means x>-1 as the all the values where 0<x<1 does not satisfy the equation.

Kudos [?]: 17 [0], given: 64

Intern
Joined: 31 Mar 2013
Posts: 8

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

Location: India
GMAT 1: 750 Q50 V40
GPA: 3.12

### Show Tags

20 May 2017, 09:13
I think this the explanation isn't clear enough, please elaborate. I understood the algebraic method but I had a doubt. An inequality problem can be solved by plugging in values also right? If we try to plug-in values into option A, the -0.5 value which we arrive by solving algebraically, will go against the statement itself. Is it that we can't solve some inequality problems by plugging in?

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

Math Expert
Joined: 02 Sep 2009
Posts: 41886

Kudos [?]: 128668 [0], given: 12181

### Show Tags

20 May 2017, 09:37
vil1 wrote:
I think this the explanation isn't clear enough, please elaborate. I understood the algebraic method but I had a doubt. An inequality problem can be solved by plugging in values also right? If we try to plug-in values into option A, the -0.5 value which we arrive by solving algebraically, will go against the statement itself. Is it that we can't solve some inequality problems by plugging in?

I'll try again.

The question asks if $$-1 \lt x \lt 0$$ or $$x \gt 1$$, then which of the following must be true.

Since $$-1 \lt x \lt 0$$ or $$x \gt 1$$, then it must be true to say about x that x > -1.

For example, x can be, among many other values, -0.9, -0.89292838, -0.76539, -0,5, ... (because $$-1 \lt x \lt 0$$) as well as x can be 3, $$\pi$$, 4.17, $$\sqrt{71}$$, ... (because $$x \gt 1$$). Any of them is greater than -1. For ANY possible x, so for ANY x from $$-1 \lt x \lt 0$$ and $$x \gt 1$$, it will be true to say that x is greater than -1.

Option A, which says that $$x \gt 1$$, is NOT always true because if x is from $$-1 \lt x \lt 0$$, say if x is -0.14, then $$x \gt 1$$ will NOT be true.

Hope it's clear.
_________________

Kudos [?]: 128668 [0], given: 12181

Intern
Status: Striving to get that elusive 740
Joined: 04 Jun 2017
Posts: 48

Kudos [?]: 11 [0], given: 24

GMAT 1: 690 Q49 V35
GPA: 3.7
WE: Analyst (Consulting)

### Show Tags

10 Jun 2017, 12:04
I think this the explanation isn't clear enough, please elaborate. what if we square both sides.

We get x2/x2<x2
x2<x4
x4-x2>0
x2(x2-1)>0
x2>0 or x2>1

this gives us
x>0 or |x|>1

Hence D. Am i missing something here?

Need the expert comment from Bunuel. Thanks

Kudos [?]: 11 [0], given: 24

Intern
Joined: 30 Mar 2015
Posts: 1

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

### Show Tags

12 Jul 2017, 19:01
I think this is a poor-quality question and the explanation isn't clear enough, please elaborate. what if x=0.5?

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

Re M09-22   [#permalink] 12 Jul 2017, 19:01

Go to page    1   2    Next  [ 35 posts ]

Display posts from previous: Sort by

# M09-22

Moderators: Bunuel, Vyshak

 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®.