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If 0 < x < 1, which of the following inequalities must be

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If 0 < x < 1, which of the following inequalities must be [#permalink] New post 08 Oct 2008, 05:48
00:00
A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

55% (01:55) correct 45% (00:44) wrong based on 43 sessions
If 0 < x < 1, which of the following inequalities must be true ?

I. x^5 < x^3
II. x^4 + x^5 < x^3 + x^2
III. x^4 - x^5 < x^2 - x^3

A. None
B. I only
C. II only
D. I and II only
E. I, II and III
[Reveal] Spoiler: OA

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Re: If 0 < x < 1, which of the following inequalities must be [#permalink] New post 08 Oct 2008, 06:55
amitdgr wrote:
If 0<x<1 , which of the following inequalities must be true ?

I. x^5 < x^3
II. x^4 + x^5 < x^3 + x^2
III. x^4 - x^5 < x^2 - x^3

* None
* I only
* II only
* I and II only
* I,II and III


The answer is E

I: x^5-x^3 = x^3*(x-1)*(x+1) < 0 since x > 0 and x < 1
II: x^4+x^5-x^3-x^2 = x^2*(x+1)*(x-1)*(x+1) < 0 for same reasons
III: x^4-x^5-x^2+x^3 = x^2*(1-x)*(x-1)*(x+1) = -x^2*(x-1)^2*(x+1) < 0
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Re: If 0 < x < 1, which of the following inequalities must be [#permalink] New post 08 Oct 2008, 07:36
My two cents:

I beeter if you divided by x^2: x^2+x^3<x+1 -> always
II better if you divided by x^2: x^2-x^3<1-x^2 always
III dividing by x^2: x^2-x^3<1-x^2
if x=0.1 thus 0.01-0.001<1-0.01 always
if x=0.9 thus 0.81-0.729<1-0.9 // 0.081<1 always

E

OA?

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Re: If 0 < x < 1, which of the following inequalities must be [#permalink] New post 08 Oct 2008, 07:47
OA is E. Thanks guys :) I missed the third equation !!
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Re: If 0 < x < 1, which of the following inequalities must be [#permalink] New post 08 Oct 2008, 08:01
I also believe the answer is E since even the third equation,
x^4 - x^5 < x^2 - x^3
= x^4 + x^3 < x^2 + x^5
=x^2 + x < 1+x^3
which is true since the value on the right hand side will be greater than 1.

thanks
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Re: If 0 < x < 1, which of the following inequalities must be [#permalink] New post 08 Oct 2008, 09:33
i will also go with E

raising a decimal number to any power reduces the value of the number.
so .1^2=.01
and .1^3=.001

can be solved by picking numbers
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Re: If 0 < x < 1, which of the following inequalities must be [#permalink] New post 10 Sep 2014, 07:17
People, please throw some more light!!
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Re: If 0 < x < 1 , which of the following inequalities must be [#permalink] New post 10 Sep 2014, 07:52
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scofield1521 wrote:
People, please throw some more light!!


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

I. x^5 < x^3
II. x^4 + x^5 < x^3 + x^2
III. x^4 - x^5 < x^2 - x^3

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

If 0 < x < 1, then x > x^2 > x^3 > x^4 > x^5 ...

I. x^5 < x^3. True.

II. x^4 + x^5 < x^3 + x^2. Each term on the left hand side is less than each term on the right hand side, thus LHS < RHS.

III. x^4 - x^5 < x^2 - x^3 --> x^4(1 - x) < x^2(1 - x). Since 0 < x < 1, then 1 - x > 0, so we can reduce by it: x^4 < x^2. True.

Answer: E.

Hope it's clear.
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Re: If 0 < x < 1, which of the following inequalities must be [#permalink] New post 10 Sep 2014, 08:15
Thanks bunuel, Its much clear now.

Bunuel wrote:
scofield1521 wrote:
People, please throw some more light!!


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

I. x^5 < x^3
II. x^4 + x^5 < x^3 + x^2
III. x^4 - x^5 < x^2 - x^3

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

If 0 < x < 1, then x > x^2 > x^3 > x^4 > x^5 ...

I. x^5 < x^3. True.

II. x^4 + x^5 < x^3 + x^2. Each term on the left hand side is less than each term on the right hand side, thus LHS < RHS.

III. x^4 - x^5 < x^2 - x^3 --> x^4(1 - x) < x^2(1 - x). Since 0 < x < 1, then 1 - x > 0, so we can reduce by it: x^4 < x^2. True.

Answer: E.

Hope it's clear.
Re: If 0 < x < 1, which of the following inequalities must be   [#permalink] 10 Sep 2014, 08:15
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