It is currently 28 Jun 2017, 09:17

# Live Now:

### 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 is positive, which of the following could be correct ordering of

Author Message
TAGS:

### Hide Tags

Intern
Joined: 18 Jun 2012
Posts: 39
Re: If x is positive, which of the following could be correct ordering of [#permalink]

### Show Tags

28 Jul 2012, 20:24
The question asked "what could be the correct ordering" means it asked for the possibilities.
What is question asked "what must be the correct ordering" ? In that case would we be required to choose an option which is true for all scenarios ?
Director
Joined: 22 Mar 2011
Posts: 612
WE: Science (Education)
Re: If x is positive, which of the following could be correct ordering of [#permalink]

### Show Tags

29 Jul 2012, 01:37
5
KUDOS
Please, refer to the attached drawing, in which the three graphs $$y=1/x,$$ $$y=2x,$$ and $$y=x^2$$ are depicted for $$x>0$$.
The exact values for A, B, and C can be worked out, but they are not important to establish the order of the three algebraic expressions.

So, the correct orderings are:
If $$x$$ between 0 and A: $$x^2<2x<1/x$$
If $$x$$ between A and B: $$x^2<1/x<2x$$
If $$x$$ between B and C: $$1/x<x^2<2x$$
If $$x$$ greater than C: $$1/x<2x<x^2$$

We can see that only the first two of the above options are listed as answers (I and II).

Attachments

3Graphs.jpg [ 15.15 KiB | Viewed 6520 times ]

_________________

PhD in Applied Mathematics
Love GMAT Quant questions and running.

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7449
Location: Pune, India
Re: If x is positive, which of the following could be correct ordering of [#permalink]

### Show Tags

29 Jul 2012, 22:58
1
KUDOS
Expert's post
smartmanav wrote:
The question asked "what could be the correct ordering" means it asked for the possibilities.
What is question asked "what must be the correct ordering" ? In that case would we be required to choose an option which is true for all scenarios ?

The ordering will be different for different values of x so the question cannot ask for a single correct ordering.
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Senior Manager Joined: 23 Oct 2010 Posts: 383 Location: Azerbaijan Concentration: Finance Schools: HEC '15 (A) GMAT 1: 690 Q47 V38 Re: If x is positive, which of the following could be correct ordering of [#permalink] ### Show Tags 08 Aug 2012, 00:50 given that x>0 1 ) x^2 < 2X<1/X lets just check x^2 < 2X => 0 <x<2 ; 2X<1/X => 0<x<1/\sqrt{2} these 2 inequities do not conradict each other. so, 1) is ok 2) x^2 <1/X< 2X check them - x^2 <1/X => 0<x<1 ; 1/X< 2X => x> 1/\sqrt{2} these 2 inequities do not conradict each other. so, 2) is ok 3) 2X< x^2 <1/X check them - 2X< x^2 => x>2 ; x^2 <1/X => x<1 these 2 inequities conradict each other. so, 3) is not ok p.s. dont know why such symbols as sqroot , fraction ets dont work. _________________ Happy are those who dream dreams and are ready to pay the price to make them come true I am still on all gmat forums. msg me if you want to ask me smth Moderator Joined: 01 Sep 2010 Posts: 3218 Re: If x is positive, which of the following could be correct ordering of [#permalink] ### Show Tags 13 Dec 2012, 07:28 picking numbers, both integers and not and only positive. In all cases only the 3 case doesn't work, so pretty fast you can reach D in this question youhave to be really comfortable with theory to solve it, otherwise is best and safe picking number . _________________ VP Joined: 09 Jun 2010 Posts: 1418 Re: If x is positive, which of the following could be correct ordering of [#permalink] ### Show Tags 14 Dec 2012, 00:30 positiveness or negativeness is important to inequalit because x is positive we can multiple both sides of inequality with x and keep the same mark.for example x^2<2x<1/x is the same as x^3<2x^2<1 (if x is negative we have to change the mark of the inequality. this question is not relevant to that cases) now solve 2 inequality independently . this can be done quick. this questions can be done in less than 3 minutes. Other method takes longer time and in fact is not good. pls, comment. _________________ visit my facebook to help me. on facebook, my name is: thang thang thang Intern Joined: 28 Aug 2012 Posts: 20 Location: United States Concentration: General Management, Leadership Schools: Thunderbird '16 GPA: 3.37 WE: Information Technology (Consulting) Re: If x is positive, which of the following could be correct ordering of [#permalink] ### Show Tags 11 May 2013, 02:05 Hey Karishma, I feel below highlighted part is not correct. Please check. If I am wrong, please explain. thanks. VeritasPrepKarishma wrote: Vavali wrote: If x is positive, which of the following could be correct ordering of $$\frac{1}{x}$$, $$2x$$, and $$x^2$$? (I) $$x^2 < 2x < \frac{1}{x}$$ (II) $$x^2 < \frac{1}{x} < 2x$$ (III) $$2x < x^2 < \frac{1}{x}$$ (a) none (b) I only (c) III only (d) I and II (e) I, II and III Let's look at this question logically. There will be some key takeaways here so don't focus on the question and the (long) solution. Focus on the logic. First of all, we are just dealing with positives so life is simpler. To compare two terms e.g. $$x^2$$ and $$2x$$, we should focus on the points where they are equal. $$x^2 = 2x$$ holds when $$x = 2$$. When $$x < 2, x^2 < 2x$$ When $$x > 2, x^2 > 2x$$ Similarly $$1/x = x^2$$ when $$x = 1$$ When $$x < 1, 1/x > x^2$$. When $$x > 1, 1/x > x^2$$ Are you sure this is correct? I think.. we can use x=4 here, then 1/4>16 .. which is not correct. Going on, $$1/x = 2x$$ when $$x = 1/\sqrt{2}$$ When $$x < 1/\sqrt{2}, 1/x > 2x$$ When $$x > 1/\sqrt{2}, 1/x < 2x$$ So now you know that: If $$x < 1/\sqrt{2}$$, $$1/x > 2x, 1/x > x^2$$ and $$x^2 < 2x$$ So $$x^2 < 2x < 1/x$$ is possible. If $$1/\sqrt{2} < x < 1$$ $$1/x < 2x, 1/x > x^2$$ So $$x^2 < 1/x < 2x$$ is possible. If $$x > 1$$ $$1/x < 2x, 1/x > x^2$$ So $$x^2 < 1/x < 2x$$ is possible. (Same as above) For no positive values of x is the third relation possible. Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7449 Location: Pune, India Re: If x is positive, which of the following could be correct ordering of [#permalink] ### Show Tags 11 May 2013, 03:12 yogirb8801 wrote: Hey Karishma, I feel below highlighted part is not correct. Please check. If I am wrong, please explain. thanks. VeritasPrepKarishma wrote: Vavali wrote: If x is positive, which of the following could be correct ordering of $$\frac{1}{x}$$, $$2x$$, and $$x^2$$? (I) $$x^2 < 2x < \frac{1}{x}$$ (II) $$x^2 < \frac{1}{x} < 2x$$ (III) $$2x < x^2 < \frac{1}{x}$$ (a) none (b) I only (c) III only (d) I and II (e) I, II and III Let's look at this question logically. There will be some key takeaways here so don't focus on the question and the (long) solution. Focus on the logic. First of all, we are just dealing with positives so life is simpler. To compare two terms e.g. $$x^2$$ and $$2x$$, we should focus on the points where they are equal. $$x^2 = 2x$$ holds when $$x = 2$$. When $$x < 2, x^2 < 2x$$ When $$x > 2, x^2 > 2x$$ Similarly $$1/x = x^2$$ when $$x = 1$$ When $$x < 1, 1/x > x^2$$. When $$x > 1, 1/x > x^2$$ Are you sure this is correct? I think.. we can use x=4 here, then 1/4>16 .. which is not correct. Going on, $$1/x = 2x$$ when $$x = 1/\sqrt{2}$$ When $$x < 1/\sqrt{2}, 1/x > 2x$$ When $$x > 1/\sqrt{2}, 1/x < 2x$$ So now you know that: If $$x < 1/\sqrt{2}$$, $$1/x > 2x, 1/x > x^2$$ and $$x^2 < 2x$$ So $$x^2 < 2x < 1/x$$ is possible. If $$1/\sqrt{2} < x < 1$$ $$1/x < 2x, 1/x > x^2$$ So $$x^2 < 1/x < 2x$$ is possible. If $$x > 1$$ $$1/x < 2x, 1/x > x^2$$ So $$x^2 < 1/x < 2x$$ is possible. (Same as above) For no positive values of x is the third relation possible. That is a typo. If you notice, for every case, the relation is opposite on the opposite sides of the equality value. So the relation that holds in x < 1 will be opposite to the relation that holds when x > 1. That did mess up the entire explanation. Good you pointed it out. I have edited the original post. _________________ 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: 39745
Re: If x is positive, which of the following could be correct ordering of [#permalink]

### Show Tags

05 Jul 2013, 02:26
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

_________________
Intern
Joined: 15 Jul 2013
Posts: 1
Re: If x is positive, which of the following could be correct ordering of [#permalink]

### Show Tags

15 Sep 2013, 10:47
1
This post was
BOOKMARKED
lylya4 wrote:
Vavali wrote:
If x is positive, which of the following could be correct ordering of 1/x, 2x, and x^2?

(I) X^2 < 2x < 1/x
(II) x^2 < 1/x < 2x
(III) 2x < x^2 < 1/x

(a) none
(b) I only
(c) III only
(d) I and II
(e) I, II and III

could be correct ordering

So if we can find any example that satisfy the inequation, that statement will be correct

(I) x = 0.1 => 0.01 < 0.2 < 10
(II) x= 1/2 => 1/4 < 1/2 < 1

(III)
2x < x^2 <=> x ( 2 -x) < 0, x > 0 then x > 2

with x > 2 ==> x^2 < 1/x <=> x^3 < 1 <=> x < 1

So (III) can't happen

if x = 1/2, 1/x = 2 .. wrong example used
Senior Manager
Joined: 07 Sep 2010
Posts: 327
Re: If x is positive, which of the following could be correct ordering of [#permalink]

### Show Tags

02 Oct 2013, 09:40
Hello Bunuel,
Can you show us graphical approach to this question.
I was able to draw the graph for all three equations and intersection points, however, I was not able to negate the third ordering. Would you please help me out.

Thanks
imhimanshu
P.S - How can I post graphs here.
_________________

+1 Kudos me, Help me unlocking GMAT Club Tests

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7449
Location: Pune, India
Re: If x is positive, which of the following could be correct ordering of [#permalink]

### Show Tags

02 Oct 2013, 21:21
1
KUDOS
Expert's post
imhimanshu wrote:
Hello Bunuel,
Can you show us graphical approach to this question.
I was able to draw the graph for all three equations and intersection points, however, I was not able to negate the third ordering. Would you please help me out.

Thanks
imhimanshu
P.S - How can I post graphs here.

Here is the graph:
Attachment:

Ques3.jpg [ 11.4 KiB | Viewed 4695 times ]

III. 2x < x^2 < 1/x

For 2x to be less than x^2, the graph of 2x should lie below the graph of x^2. This happens when the graph of 2x is the red line.
For x^2 to be less than 1/x at the same time, the graph of x^2 should lie below the graph of 1/x in the region of the red line. But in the region of the red line, the graph of x^2 is never below the graph of 1/x. It will never be because graph of 1/x is going down toward y = 0 while graph of x^2 is going up toward y = infinity.
Hence this inequality will not hold for any region.
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Intern Joined: 09 Dec 2013 Posts: 31 Re: If x is positive, which of the following could be correct ordering of [#permalink] ### Show Tags 22 Feb 2014, 08:48 I picked numbers: 1/2, 1, 3/2, 2, 3 However, it didn occur to me that I must look something like 0.9. Request experts to help me understand the logic behind picking such numbers. Have my actual GMAT in 10 days, any help would be immensely valuable! Thanks! Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7449 Location: Pune, India Re: If x is positive, which of the following could be correct ordering of [#permalink] ### Show Tags 24 Feb 2014, 02:12 1 This post received KUDOS Expert's post 1 This post was BOOKMARKED abdb wrote: I picked numbers: 1/2, 1, 3/2, 2, 3 However, it didn occur to me that I must look something like 0.9. Request experts to help me understand the logic behind picking such numbers. Have my actual GMAT in 10 days, any help would be immensely valuable! Thanks! I have answered your query using this very question here: http://www.veritasprep.com/blog/2013/05 ... on-points/ _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

Veritas Prep Reviews

Senior Manager
Joined: 10 Mar 2013
Posts: 277
GMAT 1: 620 Q44 V31
GMAT 2: 690 Q47 V37
GMAT 3: 610 Q47 V28
GMAT 4: 700 Q50 V34
GMAT 5: 700 Q49 V36
GMAT 6: 690 Q48 V35
GMAT 7: 750 Q49 V42
GMAT 8: 730 Q50 V39
Re: If x is positive, which of the following could be correct ordering of [#permalink]

### Show Tags

30 Jul 2014, 20:40
What a horrific problem?! I think a fast and simple way is to graph all functions and compare all vertically. Doing this, it can be easy to see that III is impossible.
Manager
Joined: 17 Apr 2013
Posts: 66
Location: United States
Concentration: Other, Finance
Schools: SDSU '16
GMAT 1: 660 Q47 V34
GPA: 2.76
WE: Analyst (Real Estate)
Re: If x is positive, which of the following could be correct ordering of [#permalink]

### Show Tags

13 Aug 2014, 11:39
lylya4 wrote:
Vavali wrote:
If x is positive, which of the following could be correct ordering of 1/x, 2x, and x^2?

(I) X^2 < 2x < 1/x
(II) x^2 < 1/x < 2x
(III) 2x < x^2 < 1/x

(a) none
(b) I only
(c) III only
(d) I and II
(e) I, II and III

could be correct ordering

So if we can find any example that satisfy the inequation, that statement will be correct

(I) x = 0.1 => 0.01 < 0.2 < 10
(II) x= 1/2 => 1/4 < 1/2 < 1

(III)
2x < x^2 <=> x ( 2 -x) < 0, x > 0 then x > 2

with x > 2 ==> x^2 < 1/x <=> x^3 < 1 <=> x < 1

So (III) can't happen

double check your calculation for x= 1/2 the equation won't hold true
1/0.5 = 2
_________________

Please +1 KUDO if my post helps. Thank you.

BSchool Forum Moderator
Joined: 28 Nov 2014
Posts: 912
Concentration: Strategy
Schools: Fisher '19 (M)
GPA: 3.71
Re: If x is positive, which of the following could be correct ordering of [#permalink]

### Show Tags

11 Aug 2015, 00:41
This is how I approached it

Try breaking the eqn and solve

Eq 1) x^2 < 2x < 1/x
x^2 < 2x and 2x < 1/x
or, x < 2 and x^2 < 1/2 (As X is positive, we can multiply)
x < ( 1/root 2 ) satisfies both eqn.

2) x^2 < 1/x < 2x
x^2 < 1/x and 1/x < 2x
or, x^3 < 1 and x2 > 1/2
1 > x > ( 1/root 2 ) satisfies both eqn (Eg: Root 3 / 2 )

3) 2x < x^2 < 1/x
2x < x^2 and x^2 < 1/x
or, 2 < x and x^3 < 1
Not possible

So, 4) I and 2 only

Thanks
Director
Joined: 10 Mar 2013
Posts: 597
Location: Germany
Concentration: Finance, Entrepreneurship
GMAT 1: 580 Q46 V24
GPA: 3.88
WE: Information Technology (Consulting)
Re: If x is positive, which of the following could be correct ordering of [#permalink]

### Show Tags

29 Dec 2015, 16:01
Vavali wrote:
If x is positive, which of the following could be correct ordering of $$\frac{1}{x}$$, $$2x$$, and $$x^2$$?

(I) $$x^2 < 2x < \frac{1}{x}$$
(II) $$x^2 < \frac{1}{x} < 2x$$
(III) $$2x < x^2 < \frac{1}{x}$$

(a) none
(b) I only
(c) III only
(d) I and II
(e) I, II and III

first, let's get rid of at least one x in the given expressions, as x is positive just multiply the expressions by x and we'll get
(I) $$x^3 < 2x^2 < 1$$
as $$2x^2$$ < 1 we must pick a value which is < 1, let's pick 1/2 and it works. COULD BE

(II) $$x^3 < 1< 2x^2$$
Here we can see that $$x^3$$<1 so we must pick a value <1 BUT which will make $$2x^2$$ >1 if possible. Let's pick 0.9 and it works also here. COULD BE

(III) $$2x^2 < x^3 < 1$$
We must pick a value < 1 BUT as we've already seen, if we pick a fraction < 1 we cannot make $$2x^2 < x^3$$, in the above cases it $$2x^2 was > x^3$$ each time we picked a fraction < 1

Hope it helps.
_________________

When you’re up, your friends know who you are. When you’re down, you know who your friends are.

800Score ONLY QUANT CAT1 51, CAT2 50, CAT3 50
GMAT PREP 670
MGMAT CAT 630
KAPLAN CAT 660

Intern
Joined: 04 Sep 2015
Posts: 15
WE: Asset Management (Investment Banking)
Re: If x is positive, which of the following could be correct ordering of [#permalink]

### Show Tags

29 Dec 2015, 22:13
Alternatively, we can just divide each term by x (since x > 0) and arrive at: 1/x^2 , 2 and x
Much easier to pick number this way :D
Intern
Joined: 04 Oct 2015
Posts: 2
Re: If x is positive, which of the following could be correct ordering of [#permalink]

### Show Tags

03 Jan 2016, 03:02
Hi Experts,
In solving x^2<2x=>x^2-2x<0=>x(x-2)<0=>x<0 or x<2

why did we ignore x<0 is it bcz we are told in question that x is positive number...it is so then why can't it be ignored in statement 3...btw I understood the number picking methodology, just little bit curious...Thanks in advance.
Re: If x is positive, which of the following could be correct ordering of   [#permalink] 03 Jan 2016, 03:02

Go to page   Previous    1   2   3    Next  [ 48 posts ]

Similar topics Replies Last post
Similar
Topics:
12 Which of the following could be the greatest common factor of positive 10 18 Jun 2017, 04:17
7 If x is positive, which of the following could be correct ordering of 13 18 Jun 2017, 02:10
6 If x is a positive integer, which of the following could NOT be the sq 7 30 May 2016, 04:57
31 If x is positive, which of the following could be the 15 16 Aug 2016, 02:27
1 If = 3, which of the following could be the value of x – 4? 9 08 Oct 2016, 14:26
Display posts from previous: Sort by