Last visit was: 12 Jul 2024, 20:39 It is currently 12 Jul 2024, 20:39
Toolkit
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
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

# Which of the following is a value of x for which x^11-x^3>0

SORT BY:
Tags:
Show Tags
Hide Tags
Director
Joined: 14 Jul 2010
Status:No dream is too large, no dreamer is too small
Posts: 959
Own Kudos [?]: 5008 [51]
Given Kudos: 690
Concentration: Accounting
Math Expert
Joined: 02 Sep 2009
Posts: 94302
Own Kudos [?]: 640198 [13]
Given Kudos: 84576
Math Expert
Joined: 02 Sep 2009
Posts: 94302
Own Kudos [?]: 640198 [6]
Given Kudos: 84576
General Discussion
Manager
Joined: 26 Feb 2012
Posts: 78
Own Kudos [?]: 109 [1]
Given Kudos: 56
Location: India
Concentration: General Management, Finance
WE:Engineering (Telecommunications)
Re: Which of the following is a value of x for which x^11-x^3>0 [#permalink]
1
Kudos
Bunuel wrote:
Baten80 wrote:
Which of the following is a value of x for which x^11-x^3>0
A. -2
B. -1
C. -1/2
D. 1/2
E. 1

$$x^{11}>x^3$$ to hold true either $$-1<x<0$$ or $$x>1$$. Only -1/2 is in either of the range.

Hi Bununel
Could you explain how to proceed as i stuck up after few steps???

X^11-X^3>0
=>X^3(X^8-1)>0
=>Now X has 2 solution
such as X^3>0 then X>0...1
Now X^8-1>0
=>X^8>=1
=>X>1.....2

We got 2 solution in this case I think D satisfies

Help...........

Rgds
Prasannajeet
Retired Moderator
Joined: 17 Sep 2013
Posts: 277
Own Kudos [?]: 1239 [0]
Given Kudos: 139
Concentration: Strategy, General Management
GMAT 1: 730 Q51 V38
WE:Analyst (Consulting)
Re: Which of the following is a value of x for which x^11-x^3>0 [#permalink]
Why go into such depths..

x^11 > x^3 in absolute terms for -ve or +ve value of x ...the same is reversed when x is replaced by 1/x
so options with either a positive value of x^11 or a negative value of 1/x^11..
we see the second case in option C and the question is solved in under 10 secs
Tutor
Joined: 16 Oct 2010
Posts: 15105
Own Kudos [?]: 66592 [4]
Given Kudos: 436
Location: Pune, India
Which of the following is a value of x for which x^11-x^3>0 [#permalink]
2
Kudos
2
Bookmarks
Baten80 wrote:
Which of the following is a value of x for which x^11-x^3>0

A. -2
B. -1
C. -1/2
D. 1/2
E. 1

The analysis of $$x^{11} > x^3$$ will be similar to that of $$x^3 > x$$. We know the ranges where this holds: x > 1 or -1 < x < 0.

Hence for x = -1/2, this will hold.

Originally posted by KarishmaB on 13 Mar 2014, 21:58.
Last edited by KarishmaB on 17 Oct 2022, 00:36, edited 1 time in total.
Manager
Joined: 14 Jan 2013
Posts: 114
Own Kudos [?]: 1549 [0]
Given Kudos: 30
Concentration: Strategy, Technology
GMAT Date: 08-01-2013
GPA: 3.7
WE:Consulting (Consulting)
Re: Which of the following is a value of x for which x^11-x^3>0 [#permalink]
Bunuel,

Thus we have that x^{11}>x^3 when x>1 or -1<x<1.

From Red part, why can't x=1/2?...

What am I missing here
Tutor
Joined: 16 Oct 2010
Posts: 15105
Own Kudos [?]: 66592 [2]
Given Kudos: 436
Location: Pune, India
Re: Which of the following is a value of x for which x^11-x^3>0 [#permalink]
2
Kudos
Mountain14 wrote:
Bunuel,

Thus we have that x^{11}>x^3 when x>1 or -1<x<1.

From Red part, why can't x=1/2?...

What am I missing here

Bunuel wrote:
$$x^{11}>x^3$$ to hold true either $$-1<x<0$$ or $$x>1$$. Only -1/2 is in either of the range.

This is the range given by Bunuel. $$-1<x<0$$ or $$x>1$$
So x cannot be 1/2.

The post you are quoting has a typo. In the second part, since x^3 < 0, we get x < 0.
So when you get -1 < x< 1, the only possible range is -1 < x< 0
Intern
Joined: 13 Dec 2013
Posts: 32
Own Kudos [?]: 21 [0]
Given Kudos: 21
GPA: 2.71
Re: Which of the following is a value of x for which x^11-x^3>0 [#permalink]
Baten80 wrote:
Which of the following is a value of x for which x^11-x^3>0

A. -2
B. -1
C. -1/2
D. 1/2
E. 1

plugged in and worked from wrong to right...seemed to work
GMAT Club Legend
Joined: 19 Dec 2014
Status:GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Posts: 21835
Own Kudos [?]: 11773 [4]
Given Kudos: 450
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Re: Which of the following is a value of x for which x^11-x^3>0 [#permalink]
4
Kudos
Hi All,

Since the answer choices are numbers, we can TEST THE ANSWERS to find the one that "fits." This approach is made all the easier since the answers are so simple. There are also some great Number Property shortcuts that we can take advantage of.

So, which answer would fit X^11 - X^3 > 0?

Since X^11 and X^3 both have odd exponents, plugging in 1 or -1 would yield the same result, so X^11 - X^3 = 0..... NOT > 0 like we need. Eliminate B and E.

With -2, X^11 will be CONSIDERABLY MORE NEGATIVE than X^3, so X^11 - X^3 < 0. Eliminate A.

With 1/2, X^11 will be CONSIDERABLY SMALLER than X^3, so X^11 - X^3 < 0. Eliminate D.

GMAT assassins aren't born, they're made,
Rich
Manager
Joined: 15 Aug 2014
Status:Always try to face your worst fear because nothing GOOD comes easy. You must be UNCOMFORTABLE to get to your COMFORT ZONE
Posts: 221
Own Kudos [?]: 562 [3]
Given Kudos: 470
Concentration: Marketing, Technology
GMAT 1: 570 Q44 V25
GMAT 2: 600 Q48 V25
WE:Information Technology (Consulting)
Re: Which of the following is a value of x for which x^11-x^3>0 [#permalink]
2
Kudos
1
Bookmarks
$$x^{11}>x^3$$ --> $$x^{3}(x^8-1)>0$$. So, far correct.

Next, for $$x^{3}(x^8-1)$$ to be positive $$x^3$$and $$x^8-1$$ must have the same sign, so both of them must be positive or both of them must be negative.

$$x^3>0$$ and $$x^8-1>0$$:
$$x^3>0$$ gives $$x>0$$;
$$x^8-1>0$$ --> $$x^8>1$$ --> $$x<-1$$ or $$x>1$$.
Both to hold true $$x$$ must be greater than 1. So, $$x^3>0$$ and $$x^8-1>0$$ when $$x>1$$.

$$x^3<0$$ and $$x^8-1<0$$:
$$x^3<0$$ gives $$x<0$$;
$$x^8-1<0$$ --> $$x^8<1$$ --> $$-1<x<1$$.
Both to hold true $$x$$ must be from -1 to 1. So, $$x^3<0$$ and $$x^8-1<0$$ when $$-1<x<1$$.

Thus we have that $$x^{11}>x^3$$ when $$x>1$$ or $$-1<x<1$$.

Theory on Inequalities:
x2-4x-94661.html#p731476
inequalities-trick-91482.html
data-suff-inequalities-109078.html
range-for-variable-x-in-a-given-inequality-109468.html
everything-is-less-than-zero-108884.html
graphic-approach-to-problems-with-inequalities-68037.html

All DS Inequalities Problems to practice: search.php?search_id=tag&tag_id=184
All PS Inequalities Problems to practice: search.php?search_id=tag&tag_id=189

700+ Inequalities problems: inequality-and-absolute-value-questions-from-my-collection-86939.html

Hope this helps.[/quote]

Can Someone explain how the below are deduced.

X^8>1-->X<-1 OR X>1

X^8<1--->-1<X<1
RC & DI Moderator
Joined: 02 Aug 2009
Status:Math and DI Expert
Posts: 11468
Own Kudos [?]: 34263 [1]
Given Kudos: 322
Re: Which of the following is a value of x for which x^11-x^3>0 [#permalink]
1
Kudos
smartguy595 wrote:

Can Someone explain how the below are deduced.

X^8>1-->X<-1 OR X>1

X^8<1--->-1<X<1

Hi,
when you raise anything less than 1 to an integer power, it becomes smaller as the power increases..
example 1/2.. 1/2 ^2=1/4.. 1/2^3=1/8...

lets see the two inequalities now..

X^8>1..
since the power of x is even number 8, x^8 will always be positive..
so if x is any negative integer less than -1, x^8 will >1..
example x=-2, (-2)^8 will be greater than 1..
but as we have seen earlier anything less than 1 when raised to a power will become even smaller, so it will always be<1..
that is why x cannot take any value between -1 and 1, both inclusive, for x^8 to be >1..
so x<-1 or x>1..

x^8<1..
Opposite of the above , for x^8 to be less than 1, x has to be between 1 and -1..

hope it helps
Manager
Joined: 15 Aug 2014
Status:Always try to face your worst fear because nothing GOOD comes easy. You must be UNCOMFORTABLE to get to your COMFORT ZONE
Posts: 221
Own Kudos [?]: 562 [0]
Given Kudos: 470
Concentration: Marketing, Technology
GMAT 1: 570 Q44 V25
GMAT 2: 600 Q48 V25
WE:Information Technology (Consulting)
Re: Which of the following is a value of x for which x^11-x^3>0 [#permalink]
chetan2u wrote:
smartguy595 wrote:

Can Someone explain how the below are deduced.

X^8>1-->X<-1 OR X>1

X^8<1--->-1<X<1

Hi,
when you raise anything less than 1 to an integer power, it becomes smaller as the power increases..
example 1/2.. 1/2 ^2=1/4.. 1/2^3=1/8...

lets see the two inequalities now..

X^8>1..
since the power of x is even number 8, x^8 will always be positive..
so if x is any negative integer less than -1, x^8 will >1..
example x=-2, (-2)^8 will be greater than 1..
but as we have seen earlier anything less than 1 when raised to a power will become even smaller, so it will always be<1..
that is why x cannot take any value between -1 and 1, both inclusive, for x^8 to be >1..
so x<-1 or x>1..

x^8<1..
Opposite of the above , for x^8 to be less than 1, x has to be between 1 and -1..

hope it helps

yes it is 100% clear.. Thank you so much
Intern
Joined: 11 Oct 2012
Posts: 29
Own Kudos [?]: 25 [0]
Given Kudos: 74
GMAT 1: 610 Q42 V32
Re: Which of the following is a value of x for which x^11-x^3>0 [#permalink]
VeritasPrepKarishma wrote:
Baten80 wrote:
Which of the following is a value of x for which x^11-x^3>0

A. -2
B. -1
C. -1/2
D. 1/2
E. 1

The analysis of $$x^{11} > x^3$$ will be similar to that of $$x^3 > x$$. We know the ranges where this holds: x > 1 or -1 < x < 0.

Hence for x = -1/2, this will hold.

For a question discussing ranges and the properties of numbers in these ranges, check: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/08 ... -question/

Hi Karishma ,
Can you kindly explain , how $$x^{11} > x^3$$ will be similar to that of $$x^3 > x$$ ?
Tutor
Joined: 16 Oct 2010
Posts: 15105
Own Kudos [?]: 66592 [3]
Given Kudos: 436
Location: Pune, India
Re: Which of the following is a value of x for which x^11-x^3>0 [#permalink]
2
Kudos
1
Bookmarks
AndyNeedsGMAT wrote:
VeritasPrepKarishma wrote:
Baten80 wrote:
Which of the following is a value of x for which x^11-x^3>0

A. -2
B. -1
C. -1/2
D. 1/2
E. 1

The analysis of $$x^{11} > x^3$$ will be similar to that of $$x^3 > x$$. We know the ranges where this holds: x > 1 or -1 < x < 0.

Hence for x = -1/2, this will hold.

For a question discussing ranges and the properties of numbers in these ranges, check: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/08 ... -question/

Hi Karishma ,
Can you kindly explain , how $$x^{11} > x^3$$ will be similar to that of $$x^3 > x$$ ?

When do the relations behave differently? They are different in case of even-odd powers.

A higher odd power is greater than a lower odd power when x > 1 or -1 < x < 0
So in these ranges, x^3 > x, x^5 > x, x^9 > x^5 etc all hold.

A higher even power is greater than a lower even power when x > 1 or x < -1
So in these ranges, x^8 > x^2, x^4 > x^2 etc all hold

Similarly, a higher even power is greater than a lower odd power when x > 1 or x < 0
So in these ranges, x^2 > x, x^4 > x^3 etc all hold
Intern
Joined: 11 Oct 2012
Posts: 29
Own Kudos [?]: 25 [0]
Given Kudos: 74
GMAT 1: 610 Q42 V32
Re: Which of the following is a value of x for which x^11-x^3>0 [#permalink]
Hey Karishma - Thanks a lot for clarifying my doubts.
+1 kudos
Manager
Joined: 24 May 2013
Posts: 56
Own Kudos [?]: 147 [1]
Given Kudos: 99
Which of the following is a value of x for which x^11-x^3>0 [#permalink]
1
Kudos
X^11-X^3>0
=>X^3(X^8-1)>0
=>x^3(x^4-1)(x^4+1)>0
=>x^3(x^2-1)(x^2+1)(x^4+1)>0
=>x^3(x-1)(x+1)(x^2+1)(x^4+1)>0

plotting the three roots -1, 0, 1 along with the signs of the inequality in the respective zones:
only -1<x<0 and x>1 satisfy the inequality
Attachments

x11-x3 inequality.png [ 5.36 KiB | Viewed 15113 times ]

CEO
Joined: 23 Feb 2015
Posts: 2512
Own Kudos [?]: 2137 [0]
Given Kudos: 1978
Concentration: Finance, Technology
Which of the following is a value of x for which x^11-x^3>0 [#permalink]
Bunuel wrote:
prasannajeet wrote:
Bunuel wrote:

$$x^{11}>x^3$$ to hold true either $$-1<x<0$$ or $$x>1$$. Only -1/2 is in either of the range.

Hi Bununel
Could you explain how to proceed as i stuck up after few steps???

X^11-X^3>0
=>X^3(X^8-1)>0
=>Now X has 2 solution
such as X^3>0 then X>0...1
Now X^8-1>0
=>X^8>=1
=>X>1.....2

We got 2 solution in this case I think D satisfies

Help...........

Rgds
Prasannajeet

$$x^{11}>x^3$$ --> $$x^{3}(x^8-1)>0$$. So, far correct.

Next, for $$x^{3}(x^8-1)$$ to be positive $$x^3$$and $$x^8-1$$ must have the same sign, so both of them must be positive or both of them must be negative.

$$x^3>0$$ and $$x^8-1>0$$:
$$x^3>0$$ gives $$x>0$$;
$$x^8-1>0$$ --> $$x^8>1$$ --> $$x<-1$$ or $$x>1$$.
Both to hold true $$x$$ must be greater than 1. So, $$x^3>0$$ and $$x^8-1>0$$ when $$x>1$$.

$$x^3<0$$ and $$x^8-1<0$$:
$$x^3<0$$ gives $$x<0$$;
$$x^8-1<0$$ --> $$x^8<1$$ --> $$-1<x<1$$.
Both to hold true $$x$$ must be from -1 to 1. So, $$x^3<0$$ and $$x^8-1<0$$ when $$-1<x<1$$.

Thus we have that $$x^{11}>x^3$$ when $$x>1$$ or $$-1<x<1$$.
/quote]
Hi Bunuel,
Hope you're well brother.
For the first green part we do not consider the first red part. But, why do we consider second red part (1) after knowing the second green part (x<0)
Thank you...
Manager
Joined: 09 Jan 2021
Posts: 71
Own Kudos [?]: 12 [0]
Given Kudos: 142
Location: India
Schools: ISB '23 (S)
GPA: 3.2
Re: Which of the following is a value of x for which x^11-x^3>0 [#permalink]
Baten80 wrote:
Which of the following is a value of x for which x^11-x^3>0

A. -2
B. -1
C. -1/2
D. 1/2
E. 1

The analysis of $$x^{11} > x^3$$ will be similar to that of $$x^3 > x$$. We know the ranges where this holds: x > 1 or -1 < x < 0.

Hence for x = -1/2, this will hold.

For a question discussing ranges and the properties of numbers in these ranges, check: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/08 ... -question/

Hey,
how can you say that x^11>x^3 is similar to x^3>X?
Intern
Joined: 16 Apr 2023
Posts: 37
Own Kudos [?]: [0]
Given Kudos: 107
Re: Which of the following is a value of x for which x^11-x^3>0 [#permalink]
KarishmaB I prefer to use the graphical method and normally always do. But how do you know to keep expanding the exponents like this. I have seen problems where you keep the higher powers, but here, clearly you needed to keep simplifying to get the third root. (e.g. if you had stopped at x^8(x^3-1) you only would have had 0 and 1 as roots

Vinayprajapati wrote:
X^11-X^3>0
=>X^3(X^8-1)>0
=>x^3(x^4-1)(x^4+1)>0
=>x^3(x^2-1)(x^2+1)(x^4+1)>0
=>x^3(x-1)(x+1)(x^2+1)(x^4+1)>0

plotting the three roots -1, 0, 1 along with the signs of the inequality in the respective zones:
only -1<x<0 and x>1 satisfy the inequality