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If 0 < x < 1, which of the following inequalities must be [#permalink ]
08 Oct 2008, 05:48

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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

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

Cheers

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Re: If 0 < x < 1, which of the following inequalities must be [#permalink ]
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 ]
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 SM

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