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# If x is positive, which of the following could be correct

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Re: If x is positive, which of the following could be correct [#permalink]

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27 Jun 2012, 14:11

Statement III says $$2x < x^2 < \frac{1}{x}$$

So, we analyze part by part of this compound inequality:

a) $$2x < x^2$$
then, $$x>2$$

b) $$x^2 < \frac{1}{x}$$

then,$$x^3<1$$ ==> $$x < 1$$

So, we have $$x>2$$ and $$x<1$$.
That's impossible! if $$x>2$$, x cannot be less than 1 at the same time.
Statement III could not be correct!

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Re: If x is positive, which of the following could be correct [#permalink]

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28 Jun 2012, 02:15
Expert's post
metallicafan wrote:

Statement III says $$2x < x^2 < \frac{1}{x}$$

So, we analyze part by part of this compound inequality:

a) $$2x < x^2$$
then, $$x>2$$

b) $$x^2 < \frac{1}{x}$$

then,$$x^3<1$$ ==> $$x < 1$$

So, we have $$x>2$$ and $$x<1$$.
That's impossible! if $$x>2$$, x cannot be less than 1 at the same time.
Statement III could not be correct!

Yes, your reasoning for option III is correct.
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Re: If x is positive, which of the following could be correct [#permalink]

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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 ?
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Re: If x is positive, which of the following could be correct [#permalink]

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29 Jul 2012, 01:37
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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).

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Re: If x is positive, which of the following could be correct [#permalink]

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29 Jul 2012, 22:58
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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.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Current Student Joined: 23 Oct 2010 Posts: 386 Location: Azerbaijan Concentration: Finance Schools: HEC '15 (A) GMAT 1: 690 Q47 V38 Followers: 21 Kudos [?]: 275 [0], given: 73 Re: If x is positive, which of the following could be correct [#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: 2992 Followers: 691 Kudos [?]: 5568 [0], given: 968 Re: If x is positive, which of the following could be correct [#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 . _________________ Intern Joined: 13 Dec 2012 Posts: 4 Followers: 0 Kudos [?]: 0 [0], given: 0 Re: If x is positive, which of the following could be correct [#permalink] ### Show Tags 13 Dec 2012, 23:41 if x = 1/2, then 1/1/2 = 2 which is greater than 2(1/2) VP Joined: 09 Jun 2010 Posts: 1315 Followers: 3 Kudos [?]: 90 [0], given: 745 Re: If x is positive, which of the following could be correct [#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. Intern Joined: 28 Aug 2012 Posts: 20 Location: United States Concentration: General Management, Leadership Schools: Thunderbird '16 GPA: 3.37 WE: Information Technology (Consulting) Followers: 1 Kudos [?]: 53 [0], given: 6 Re: If x is positive, which of the following could be correct [#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: 6761 Location: Pune, India Followers: 1877 Kudos [?]: 11556 [0], given: 219 Re: If x is positive, which of the following could be correct [#permalink] ### Show Tags 11 May 2013, 03:12 Expert's post 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

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Re: If x is positive, which of the following could be correct [#permalink]

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05 Jul 2013, 02:26
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Re: If X is positive [#permalink]

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15 Sep 2013, 10:47
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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
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Re: If x is positive, which of the following could be correct [#permalink]

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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.
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Re: If x is positive, which of the following could be correct [#permalink]

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02 Oct 2013, 21:21
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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 3441 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.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Current Student Joined: 06 Sep 2013 Posts: 2035 Concentration: Finance GMAT 1: 770 Q0 V Followers: 51 Kudos [?]: 497 [0], given: 355 Re: If x is positive, which of the following could be correct [#permalink] ### Show Tags 14 Feb 2014, 08:49 Bunuel wrote: imhimanshu wrote: Hi Bunnel, Could you please provide a reasoning to the below text... how did you find the range...Pls help The reasoning is that in these ranges x (2x), 1/x and x^2 are ordered differently: For $$x>2$$ --> $$x^2$$ has the largest value. Since no option offers this we know that $$x$$ cannot be more that 2; For $$1<x<2$$ --> $$2x$$ has the largest value, then comes $$x^2$$. Since no option offers this we know that $$x$$ cannot be from this range either; So, we are left with last range: $$0<x<1$$. In this case $$x^2$$ has the least value. Options, I and II offer this, so we should concentrate on them and test the values of x from 0 to 1. Hope it's clear. Hi Bunuel. How do you know which numbers to pick? I mean for example 0.9 for the second statement. Any clues? Thanks Cheers J Intern Joined: 09 Dec 2013 Posts: 31 Followers: 1 Kudos [?]: 36 [0], given: 8 Re: If x is positive, which of the following could be correct [#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: 6761 Location: Pune, India Followers: 1877 Kudos [?]: 11556 [1] , given: 219 Re: If x is positive, which of the following could be correct [#permalink] ### Show Tags 24 Feb 2014, 02:12 1 This post received KUDOS Expert's post 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

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Re: If x is positive, which of the following could be correct [#permalink]

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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.
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Re: If x is positive, which of the following could be correct [#permalink]

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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
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Re: If x is positive, which of the following could be correct   [#permalink] 13 Aug 2014, 11:39

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