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

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If 0<x<1, which of the following must be true? 1. [#permalink] New post   20 Aug 2006, 18:05
If 0<x<1, which of the following must be true?

1. x^5< X^3
2. X^4+ X^5 < X^3+ X ^ 2
3. X^4 - X^5 < X^3 - X ^ 2

Can some one help me with the working for option 3?



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Re: If 0<x<1, which of the following must be true? 1. [#permalink] New post   20 Aug 2006, 18:29
1. x^5< X^3
2. X^4+ X^5 < X^3+ X ^ 2
3. X^4 - X^5 < X^3 - X ^ 2

1. True : x is a positive fraction
2. True : x^2(x^3 + x^2) < x^3 + x^2
3. True : -x^2(x^3 -x^2) < x^3 - x^2 : -x^2 < 1 (x is positive)


Can you confirm the answer?
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Re: If 0<x<1, which of the following must be true? 1. [#permalink] New post   20 Aug 2006, 18:36
gk3.14

Yup you got it correct. Also I really like you solution.

Thanks
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Re: If 0<x<1, which of the following must be true? 1. [#permalink] New post   20 Aug 2006, 18:59
Only 1 and 2 must be true.

For 0<x<1 it is obvious that for every increasing power the value actually reduces..

i.e. x>x^2>x^3 .. and so on.

Hence statements 1 and 2 are always true.

For Statement III ..

X^4 - X^5 < X^3 - X ^ 2
X^4+X^2 < X^3 + X^5
X^4 + X ^ 2 < X(X^2 + x^4)

which implies that x>1 hence 3 is not possible.
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Re: If 0<x<1, which of the following must be true? 1. [#permalink] New post   20 Aug 2006, 19:06
gmatornot,

I used the same logic but OA is all three are correct.
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Re: If 0<x<1, which of the following must be true? 1. [#permalink] New post   20 Aug 2006, 20:00
2times wrote:
If 0<x<1, which of the following must be true?

1. x^5< X^3
2. X^4+ X^5 < X^3+ X ^ 2
3. X^4 - X^5 < X^3 - X^2

Can some one help me with the working for option 3?


i donot think 3 is correct. lets use plug-in with x = 1/2

LHS = X^4 - X^5
= (1/2)^4 - (1/2)^5
= (1/2)^4 (1 - 1/2)
= (1/2)^4 (1/2)
= (1/2)^5

RHS = X^3 - X^2
= (1/2)^3 - (1/2)^2
= (1/2)^2 (1/2 - 1)
= (1/2)^2 (-1/2)
= - (1/2)^3

therefore (X^4 - X^5) > (X^3 - X^2)



gk3.14 wrote:
1. x^5< X^3
2. X^4+ X^5 < X^3+ X ^ 2
3. X^4 - X^5 < X^3 - X ^ 2

1. True : x is a positive fraction
2. True : x^2(x^3 + x^2) < x^3 + x^2
3. True : -x^2(x^3 -x^2) < x^3 - x^2 : -x^2 < 1 (x is positive)Can you confirm the answer?


you cannot solve the inequality as it is. you are solving the inequality as if the given inequality is correct. also when you divide the inequality by -ves, the inequality should be fliped.
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Re: If 0<x<1, which of the following must be true? 1. [#permalink] New post   20 Aug 2006, 20:05
check this...
[/img]
Attachments

ineq.doc [61 KiB]
Downloaded 97 times

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Re: If 0<x<1, which of the following must be true? 1. [#permalink] New post   20 Aug 2006, 20:39
2times wrote:
If 0<x<1, which of the following must be true?

1. x^5< X^3
2. X^4+ X^5 < X^3+ X ^ 2
3. X^4 - X^5 < X^3 - X ^ 2

Can some one help me with the working for option 3?


1. x^5-x^3 < 0
x^3(x^2-1) < 0
since 0<x<1 --> x^3 cannot be -ve
Hence (x^2-1) < 0
--> 0<x<1 True

2. x^4(1+x) < x^2( 1+x)
x^4 < x^2
x^2(x^2 -1) < 0
--> x^2 cannot be -ve
Hence (x^2-1) < 0
--> 0<x<1 True

3. x^4(1-x) < -x^2( 1-x)
x^4 + x^2 < 0
x^2(x^2 +1) < 0
--> x^2 cannot be -ve
(x^2 +1) < 0
x^ 2 < -1 which is not possible

Hence Only 1 & 2 are True.

I dont buy the OA.

Heman
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Re: If 0<x<1, which of the following must be true? 1. [#permalink] New post   20 Aug 2006, 20:43
2times wrote:
If 0<x<1, which of the following must be true?

1. x^5< X^3
2. X^4+ X^5 < X^3+ X ^ 2
3. X^4 - X^5 < X^3 - X ^ 2


lets not go after who said what and what is OA given?

do you agree that (X^4 - X^5) is +ve and (X^3 - X^2) is -ve since x is a fraction. you tell me, which is greater? i am 100% sure that statememnt 3 is not correct as given by you.

note: make sure you posted question correctly. :wink:
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Re: If 0<x<1, which of the following must be true? 1. [#permalink] New post   20 Aug 2006, 20:58
2times wrote:
If 0<x<1, which of the following must be true?

3. X^4 - X^5 < X^3 - X ^ 2

Can some one help me with the working for option 3?


2 times I checked the attached Q
Statement 3 should be

X^4 - X^5 < x^2 - x^3
In that case
x^4(1-x) < x^2(1-x)
x^4-x^2<0
x^2( x^2-1) < 0
x^2 cannot be -ve
Hence x^2-1 < 0
which implies 0<x<1 Hence True
Hence (3) is true.
(1) & (2) are true as shown be4

Heman
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Re: If 0<x<1, which of the following must be true? 1. [#permalink] New post   20 Aug 2006, 21:13
no wonder I get so many quant questions incorrect due to silly mistakes :(
Thanks guys!
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Re: If 0<x<1, which of the following must be true? 1. [#permalink] New post   20 Aug 2006, 22:20
Ah.. Thanks Prof.. I was treating these inequalites as if they were true..
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Re: If 0<x<1, which of the following must be true? 1. [#permalink] New post   22 Aug 2006, 08:41
As originally written, only satements I and II are true.

After being edited, all three statements stand.

Just pick 2 for x if you don`t know the rule of fractional exponents.

Sometimes Kaplan strategies work (Kan Always Pick Little Abstract Numbers)
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Re: If 0<x<1, which of the following must be true? 1. [#permalink] New post   24 Aug 2006, 20:18
heman wrote:
2times wrote:
If 0<x<1, which of the following must be true?

3. X^4 - X^5 < X^3 - X ^ 2

Can some one help me with the working for option 3?


2 times I checked the attached Q
Statement 3 should be

X^4 - X^5 < x^2 - x^3
In that case
x^4(1-x) < x^2(1-x)
x^4-x^2<0
x^2( x^2-1) < 0
x^2 cannot be -ve
Hence x^2-1 < 0
which implies 0<x<1 Hence True
Hence (3) is true.
(1) & (2) are true as shown be4

Heman


Heman, I guess you went wrong in one place.
when x^4(1-x) < x^2(1-x)
You divided by (1-x) on both sides, to get x^4-x^2<0.
But you cant do this in inequalities, since you dont know what the value of 1-x is. If the value is negative the inequality sign needs to be reversed.
Hence (3) is false.
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Re: If 0<x<1, which of the following must be true? 1. [#permalink] New post   24 Aug 2006, 22:16
yessuresh wrote:
heman wrote:
2times wrote:
If 0<x<1, which of the following must be true?

3. X^4 - X^5 < X^3 - X ^ 2

Can some one help me with the working for option 3?


2 times I checked the attached Q
Statement 3 should be

X^4 - X^5 < x^2 - x^3
In that case
x^4(1-x) < x^2(1-x)
x^4-x^2<0
x^2( x^2-1) < 0
x^2 cannot be -ve
Hence x^2-1 < 0
which implies 0<x<1 Hence True
Hence (3) is true.
(1) & (2) are true as shown be4

Heman


Heman, I guess you went wrong in one place.
when x^4(1-x) < x^2(1-x)
You divided by (1-x) on both sides, to get x^4-x^2<0.
But you cant do this in inequalities, since you dont know what the value of 1-x is. If the value is negative the inequality sign needs to be reversed.
Hence (3) is false.


yessuresh

U can divide in this case since Qstem states that x is +ve 0<x<1

Heman



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Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
Where to now? Join ongoing discussions on thousands of quality questions in our Quantitative Questions Forum
Still interested in this question? Check out the "Best Topics" block above for a better discussion on this exact question, as well as several more related questions.
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Re: If 0<x<1, which of the following must be true? 1. [#permalink]
24 Aug 2006, 22:16
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