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How many of the following inequalities are possible for at least one  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 59% (01:48) correct 41% (01:52) wrong based on 315 sessions

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How many of the following inequalities are possible for at least one value of x?

I. $$x^2 > x > x^3$$

II. $$x^2 > x^3 > x$$

III. $$x^3 > x > x^2$$

A. I only
B. II only
C. I and II only
D. I, II, and III only
E. none

Originally posted by tommannanchery on 31 Jul 2016, 12:21.
Last edited by Bunuel on 03 Jul 2017, 03:23, edited 2 times in total.
Renamed the topic and edited the question.
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Re: How many of the following inequalities are possible for at least one  [#permalink]

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tommannanchery wrote:
How many of the following equalities are possible for at least one value of X?

(i) $$x^2$$ > X > $$x^3$$
(ii) $$x^2$$ > $$x^3$$ > X
(iii) $$x^3$$ > X > $$x^2$$

A. i only
B. ii only
C. i and ii only
D. i, ii, and iii only
E. none

The question is based on your understanding of how x, x^2 and x^3 behave in different regions of the number line.

x^2 > x for all negative x and when x > 1
x^3 > x for x > 1 and -1 < x < 0.

(i) $$x^2$$ > X > $$x^3$$
True for all negative x.

(ii) $$x^2$$ > $$x^3$$ > X
True for -1 < x< 0

(iii) $$x^3$$ > X > $$x^2$$
x is greater than x^2 between 0 and 1 only. But in that region x^3 is not greater than x. So this will not hold.

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Re: How many of the following inequalities are possible for at least one  [#permalink]

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tommannanchery wrote:
How many of the following equalities are possible for at least one value of X?

(i) $$x^2$$ > X > $$x^3$$
(ii) $$x^2$$ > $$x^3$$ > X
(iii) $$x^3$$ > X > $$x^2$$

A. i only
B. ii only
C. i and ii only
D. i, ii, and iii only
E. none

(1) Let X=-2 then we get a yes answer
(2) Let X=-0.5 then we get a yes
(3) none of X satisfies

Ans C

Originally posted by rohit8865 on 31 Jul 2016, 19:15.
Last edited by rohit8865 on 01 Aug 2016, 07:31, edited 2 times in total.
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Re: How many of the following inequalities are possible for at least one  [#permalink]

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Answer should be C. iii is incorrect because when x > x2, which is when 0
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Re: How many of the following inequalities are possible for at least one  [#permalink]

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tommannanchery wrote:
How many of the following equalities are possible for at least one value of X?

(i) $$x^2$$ > X > $$x^3$$
(ii) $$x^2$$ > $$x^3$$ > X
(iii) $$x^3$$ > X > $$x^2$$

A. i only
B. ii only
C. i and ii only
D. i, ii, and iii only
E. none

For (i) - Consider x=1/3. => We can have X= 2/27 which is between 1/9 and 1/27

For (ii)- Consider x= 1/2= > We can have X=0 which is less than 1/4 and 1/8.

For (iii) - Consider x= 2 => We can have X = 5 which is between 8 and 4.

Hence, we can have each of the above statements valid for different vales of x and X.

NOTE : I have solved the above question based on the assumption that x and X are two different variables.
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Re: How many of the following inequalities are possible for at least one  [#permalink]

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abhimahna wrote:
tommannanchery wrote:
How many of the following equalities are possible for at least one value of X?

(i) $$x^2$$ > X > $$x^3$$
(ii) $$x^2$$ > $$x^3$$ > X
(iii) $$x^3$$ > X > $$x^2$$

A. i only
B. ii only
C. i and ii only
D. i, ii, and iii only
E. none

For (i) - Consider x=1/3. => We can have X= 2/27 which is between 1/9 and 1/27

For (ii)- Consider x= 1/2= > We can have X=0 which is less than 1/4 and 1/8.

For (iii) - Consider x= 2 => We can have X = 5 which is between 8 and 4.

Hence, we can have each of the above statements valid for different vales of x and X.

NOTE : I have solved the above question based on the assumption that x and X are two different variables.

if the highlighted part is correct then there is no need of any option there always exist a number in between

Karishma

Please give your 2 cents also because source mentioned is veritas... Board of Directors V
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Re: How many of the following inequalities are possible for at least one  [#permalink]

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rohit8865 wrote:
abhimahna wrote:
tommannanchery wrote:
How many of the following equalities are possible for at least one value of X?

(i) $$x^2$$ > X > $$x^3$$
(ii) $$x^2$$ > $$x^3$$ > X
(iii) $$x^3$$ > X > $$x^2$$

A. i only
B. ii only
C. i and ii only
D. i, ii, and iii only
E. none

For (i) - Consider x=1/3. => We can have X= 2/27 which is between 1/9 and 1/27

For (ii)- Consider x= 1/2= > We can have X=0 which is less than 1/4 and 1/8.

For (iii) - Consider x= 2 => We can have X = 5 which is between 8 and 4.

Hence, we can have each of the above statements valid for different vales of x and X.

NOTE : I have solved the above question based on the assumption that x and X are two different variables.

if the highlighted part is correct then there is no need of any option there always exist a number in between

Karishma

Please give your 2 cents also because source mentioned is veritas... Correct bro. I agree to what you said. _________________
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Re: How many of the following inequalities are possible for at least one  [#permalink]

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VeritasPrepKarishma wrote:
tommannanchery wrote:
How many of the following equalities are possible for at least one value of X?

(i) $$x^2$$ > X > $$x^3$$
(ii) $$x^2$$ > $$x^3$$ > X
(iii) $$x^3$$ > X > $$x^2$$

A. i only
B. ii only
C. i and ii only
D. i, ii, and iii only
E. none

The question is based on your understanding of how x, x^2 and x^3 behave in different regions of the number line.

x^2 > x for all negative x and when x > 1
x^3 > x for x > 1 and -1 < x < 0.

(i) $$x^2$$ > X > $$x^3$$
True for all negative x.

(ii) $$x^2$$ > $$x^3$$ > X
True for -1 < x< 0

(iii) $$x^3$$ > X > $$x^2$$
x is greater than x^2 between 0 and 1 only. But in that region x^3 is not greater than x. So this will not hold.

Responding to a pm:
Quote:
I have a doubt for the second one?

-0.5 (-1/2) lies in the range -1<x<0

so if we consider x=-1/2; then x2 = -1/4 and x3 = -1/8

in this case x3>x2>x -_. so second case does not hold

Note that square of a real number is never negative.

If

$$x = -1/2,$$

$$x^2 = (-1/2)^2 = (-1/2)*(-1/2) = 1/4$$

$$x^3 = (-1/2)*(-1/2)*(-1/2) = -1/8$$

So
$$x^2 > x^3 > x$$
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Re: How many of the following inequalities are possible for at least one  [#permalink]

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tommannanchery wrote:
How many of the following equalities are possible for at least one value of X?

(i) $$x^2$$ > X > $$x^3$$
(ii) $$x^2$$ > $$x^3$$ > X
(iii) $$x^3$$ > X > $$x^2$$

A. i only
B. ii only
C. i and ii only
D. i, ii, and iii only
E. none

Bunuel: Can you please check this question? I did not find any values for which (iii) holds. Shouldn't OA be C?
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Joined: 02 Sep 2009
Posts: 58402
Re: How many of the following inequalities are possible for at least one  [#permalink]

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warriorguy wrote:
tommannanchery wrote:
How many of the following equalities are possible for at least one value of X?

(i) $$x^2$$ > X > $$x^3$$
(ii) $$x^2$$ > $$x^3$$ > X
(iii) $$x^3$$ > X > $$x^2$$

A. i only
B. ii only
C. i and ii only
D. i, ii, and iii only
E. none

Bunuel: Can you please check this question? I did not find any values for which (iii) holds. Shouldn't OA be C?

Yes. The correct answer is C, not D. Edited. Thank you.
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Re: How many of the following inequalities are possible for at least one  [#permalink]

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saichandm wrote:
Answer should be C. iii is incorrect because when x > x2, which is when 0<x<1), x3 is less than x2!!

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For a problem that hinges on a conceptual understanding of properties of numbers with exponents, plugging in is a natural first approach. So try values in meaningfully different ranges. Say x is 2: that makes
$$x^2$$=4
and $$x^3$$=8
$$So, x^3 > x^2 > x$$
which doesn't work for any of the inequalities.
Now say x is -2: that makes $$x^2$$
$$x^3$$=-8

$$So, x^2 > x > x^3$$

Having tried both negative and positive integers, consider what other ranges are left: fractions between 0 and 1 and fractions between 0 and 1. Inequality (ii) will be true for any fraction between 0 and -1, so it's possible, but inequality (iii) never works: the only numbers for which x > $$x^2$$ are fractions between 0 and 1, but for all such numbers $$x^2$$ is also greater than $$x^3$$

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