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If -1 < x < 0, which of the following must be true?

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If -1 < x < 0, which of the following must be true? [#permalink]

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If -1 < x < 0, which of the following must be true?

I. x^3 < x^2
II. x^5 < 1 – x
III. x^4 < x^2

A. I only
B. I and II only
C. II and III only
D. I and III only
E. I, II and III
[Reveal] Spoiler: OA

Last edited by Bunuel on 31 Jan 2013, 06:29, edited 2 times in total.
Renamed the topic, edited the question, moved to PS forum and edited OA.
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Re: if -1<x<o, which of following must be true? [#permalink]

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New post 31 Jan 2013, 06:28
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rakesh20j wrote:
if -1<x<o, which of following must be true?
1) x^3<x^2 2) x^5<1-x 3) x^4<x^2

A. I only B. I and II C. I II and III D. I and III E. II and III


The question should read:

If -1 < x < 0, which of the following must be true?

I. x^3 < x^2
II. x^5 < 1 – x
III. x^4 < x^2


A. I only
B. I and II only
C. II and III only
D. I and III only
E. I, II and III

I. x^3 < x^2 --> from -1 < x < 0 it follows that LHS<0<RHS, so this statement is true.

II. x^5 < 1 – x --> x(x^4+1) < 1 --> negative*positive < 0 < 1, so this statement is also true.

III. x^4 < x^2 --> reduce by x^2 (we can safely do that since from -1 < x < 0 is follows that x^2>0): x^2 < 1. Again, as -1 < x < 0, then x^2 must be less than 1. Hence, this statement is also true.

Answer: E.

Hope it's clear.
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Re: if -1<x<o, which of following must be true? [#permalink]

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New post 01 Jun 2013, 03:49
Bunuel wrote:
rakesh20j wrote:
if -1<x<o, which of following must be true?
1) x^3<x^2 2) x^5<1-x 3) x^4<x^2

A. I only B. I and II C. I II and III D. I and III E. II and III


The question should read:

If -1 < x < 0, which of the following must be true?

I. x^3 < x^2
II. x^5 < 1 – x
III. x^4 < x^2


A. I only
B. I and II only
C. II and III only
D. I and III only
E. I, II and III

I. x^3 < x^2 --> from -1 < x < 0 it follows that LHS<0<RHS, so this statement is true.

II. x^5 < 1 – x --> x(x^4+1) < 1 --> negative*positive < 0 < 1, so this statement is also true.

III. x^4 < x^2 --> reduce by x^2 (we can safely do that since from -1 < x < 0 is follows that x^2>0): x^2 < 1. Again, as -1 < x < 0, then x^2 must be less than 1. Hence, this statement is also true.

Answer: E.

Hope it's clear.


Bunuel,

Did not understand the colored part. :(

As -1 < x < 0, then X will always be negative and X2 will always be positive, so how to derive that x<1.
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Re: if -1<x<o, which of following must be true? [#permalink]

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New post 01 Jun 2013, 03:57
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targetbschool wrote:
Bunuel wrote:
rakesh20j wrote:
if -1<x<o, which of following must be true?
1) x^3<x^2 2) x^5<1-x 3) x^4<x^2

A. I only B. I and II C. I II and III D. I and III E. II and III


The question should read:

If -1 < x < 0, which of the following must be true?

I. x^3 < x^2
II. x^5 < 1 – x
III. x^4 < x^2


A. I only
B. I and II only
C. II and III only
D. I and III only
E. I, II and III

I. x^3 < x^2 --> from -1 < x < 0 it follows that LHS<0<RHS, so this statement is true.

II. x^5 < 1 – x --> x(x^4+1) < 1 --> negative*positive < 0 < 1, so this statement is also true.

III. x^4 < x^2 --> reduce by x^2 (we can safely do that since from -1 < x < 0 is follows that x^2>0): x^2 < 1. Again, as -1 < x < 0, then x^2 must be less than 1. Hence, this statement is also true.

Answer: E.

Hope it's clear.


Bunuel,

Did not understand the colored part. :(

As -1 < x < 0, then X will always be negative and X2 will always be positive, so how to derive that x<1.


Consider this: \(x^4 < x^2\) holds true if \(-1<x<0\) or \(0<x<1\).

Therefore, since given that \(-1<x<0\), then \(x^4 < x^2\) must be true.

Hope it's clear.
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Re: If -1 < x < 0, which of the following must be true? [#permalink]

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Re: If -1 < x < 0, which of the following must be true? [#permalink]

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If -1 < x < 0, which of the following must be true? [#permalink]

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New post 06 Mar 2016, 04:41
If \(-1 < x < 0\), which of the following must be true?

I. \(x^3 < x^2\)
II.\(x^5 < 1 – x\)
III.\(x^4 < x^2\)

A. I only
B. I and II only
C. II and III only
D. I and III only
E. I, II and III

Last edited by Engr2012 on 06 Mar 2016, 04:46, edited 1 time in total.
Reformatted the question
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If -1 < x < 0, which of the following must be true? [#permalink]

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New post 06 Mar 2016, 04:54
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animeshk wrote:
If \(-1 < x < 0\), which of the following must be true?

I. \(x^3 < x^2\)
II.\(x^5 < 1 – x\)
III.\(x^4 < x^2\)

A. I only
B. I and II only
C. II and III only
D. I and III only
E. I, II and III


Make sure to format the question properly. When you select the tag "source-others please specify" , do mention at the end of the question.

As for the question look below.

You are told that \(-1<x<0\) ---> x is a negative fraction less -1.

Let us analyse the 3 options.

I. \(x^3 < x^2\)----> MUST BE TRUE as \(x^2\) of \('x'\) in this question >0 while \(x^3 <0\). Consider \(x=-0.5\), \(x^2=0.25 >0\) while \(x^3=-0.125 <0\)[/m].

II. \(x^5 < 1 – x\) ----> MUST BE TRUE. Odd power of a negative umber \(x\) (=\(x^5\))<0 while 1-x for a negative \(x\) >0. Consider \(x=-0.5\) ---> \(1-(-0.5)=1+0.5=1.5 >0\)

III. \(x^4 < x^2\) -----> MUST BE TRUE. For fractions -1<x<0 or 0>x>1, greater the power, smaller will be the number. Consider \(x=-0.5\), \(x^2=0.25\), \(x^4 = 0.0625\), thus\(x^4<x^2\)

Thus, I,II,III are all MUST BE TRUE ----> E is the correct answer.

Hope this helps.
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Re: If -1 < x < 0, which of the following must be true? [#permalink]

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New post 06 Mar 2016, 05:53
Expert's post
animeshk wrote:
If \(-1 < x < 0\), which of the following must be true?

I. \(x^3 < x^2\)
II.\(x^5 < 1 – x\)
III.\(x^4 < x^2\)

A. I only
B. I and II only
C. II and III only
D. I and III only
E. I, II and III


Hi,
this Qs tests the properties of numbers/ fractions between 0 and 1..

so lets first check the properties, the answer will come automatically..

1) any even power will always be greater than any odd power..
x^even>x or x^odd..

2) within even powers--
as we keep increasing power the value will keep becoming lesser..
x^4<x^2..

3) Opposite will be true of ODD powers
although the numeric value will be same as in even powers but due to a -ive sign, opposite is true..
x^3>x

4) Roots--
even roots will be imaginary..
\sqrt{x} not real
Odd roots are possible and will be lesser for lower
3rd root will be more than 5th root of x.. or 3rd root of x<x

so lets see three choices now--
I. \(x^3 < x^2\)
TRUE as per point 1) above

II.\(x^5 < 1 – x\)
x is negative so -x will be positive and 1-x will be positive too..
whereas x^5 is -ive..
hence TRUE

III.\(x^4 < x^2\)
TRUE as per point 2) above

All three are correct
ans E

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Re: If -1 < x < 0, which of the following must be true? [#permalink]

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New post 07 Mar 2016, 09:38
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Re: If -1 < x < 0, which of the following must be true?   [#permalink] 07 Mar 2016, 09:38
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