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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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31 Jan 2013, 06:02
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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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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.

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

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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.

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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01 Jun 2013, 03:57
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Expert's post
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.

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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25 Jun 2014, 05:01
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Re: If -1 < x < 0, which of the following must be true? [#permalink]

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15 Sep 2015, 04:03
Hello from the GMAT Club BumpBot!

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

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06 Mar 2016, 04:54
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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06 Mar 2016, 05:53
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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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22 Feb 2017, 00:11
rakesh20j 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

I. x^3 < x^2
---> x<1 [dividing by x^2 in both side]
---> for negative value of x is less than 1. So, true.

II. x^5 < 1 – x
---> x^5+x<1
---> negative result-negative result<1
---> negative result<1
---> So, true.

III. x^4 < x^2
---> x^2<1 [dividing by x^2 in both side]
---> for any value of x (according to question stem) it must be less than 1
---> So, true.
So, the correct choice is E to me.
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Re: If -1 < x < 0, which of the following must be true? [#permalink]

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19 Jun 2017, 15:52
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.

Hope it's clear.

Try by substituting a value in the equation.

Take x = -1/2 and cross check each option.

I. x^3 < x^2 - True
II. x^5 < 1 – x - True
III. x^4 < x^2[/b] - True

Hence, answer is E - All - I, II & III
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Re: If -1 < x < 0, which of the following must be true? [#permalink]

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20 Jun 2017, 19:18
rakesh20j 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

We can see that x is a negative number between 0 and -1. Thus, x^odd = negative and x^even = positive. So, I is true.

Furthermore, 1 - x = 1 + |x|, which is positive, so it’s greater than x^5, which is negative. So, II is also true.

Lastly, when x is raised to an even integer power, the larger the even integer, the smaller the power. For example, if x = -1/2, then x^4 = 1/16 and x^2 = 1/4. Thus, x^4 < x^2. So, III is also true .

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

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20 Jun 2017, 22:54
imo E
take a value of -0.2 then substitute in every equation .
hence all satisfy
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Re: If -1 < x < 0, which of the following must be true? [#permalink]

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21 Jun 2017, 00:40
rakesh20j 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

Here it is given that -1<x<0 so x is a -ve number greater than -1.
I. x^3 < x^2 is true . X^2 is positive, x^3 is -ve
II. x^5 < 1-x i true. X^5 is -ve , 1-x is +ve.
III. X^4 < x^2 is true. X^4 and x^2 are both +ve. Also since x has a domain (-1,0), x^4 < x^2.

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Re: If -1 < x < 0, which of the following must be true?   [#permalink] 21 Jun 2017, 00:40
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