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Re: Which of the following is a value of x for which x^11-x^3>0 [#permalink]
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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.

Answer: C.


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
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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
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Which of the following is a value of x for which x^11-x^3>0 [#permalink]
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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.
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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 :(
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Re: Which of the following is a value of x for which x^11-x^3>0 [#permalink]
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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.

Answer: C.


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
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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
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Re: Which of the following is a value of x for which x^11-x^3>0 [#permalink]
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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.

Final Answer:

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Re: Which of the following is a value of x for which x^11-x^3>0 [#permalink]
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\(x^{11}>x^3\) --> \(x^{3}(x^8-1)>0\). 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
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Re: Which of the following is a value of x for which x^11-x^3>0 [#permalink]
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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
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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
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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 \(x^3 > x\) ?
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Re: Which of the following is a value of x for which x^11-x^3>0 [#permalink]
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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
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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
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Which of the following is a value of x for which x^11-x^3>0 [#permalink]
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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
So answer is C
Attachments

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

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.

Answer: C.


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...
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Re: Which of the following is a value of x for which x^11-x^3>0 [#permalink]
VeritasKarishma 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/




Hey,
how can you say that x^11>x^3 is similar to x^3>X?
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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
So answer is C
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Re: Which of the following is a value of x for which x^11-x^3>0 [#permalink]
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