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

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Manager
Joined: 25 Jul 2006
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If 0<x<1, which of the following must be true? 1. [#permalink]  20 Aug 2006, 17:05
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?
Senior Manager
Joined: 14 Aug 2006
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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)

Manager
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Kudos [?]: 4 [0], given: 0

gk3.14

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

Thanks
Senior Manager
Joined: 14 Jul 2005
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Kudos [?]: 12 [0], given: 0

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.
Manager
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gmatornot,

I used the same logic but OA is all three are correct.
VP
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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.
Manager
Joined: 25 Jul 2006
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check this...
[/img]
Attachments

ineq.doc [61 KiB]

Manager
Joined: 20 Mar 2006
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Kudos [?]: 1 [0], given: 0

Re: GMATprep - exponents & inequalities [#permalink]  20 Aug 2006, 19:39
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.

Heman
VP
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Re: GMATprep - exponents & inequalities [#permalink]  20 Aug 2006, 19:43
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.
Manager
Joined: 20 Mar 2006
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Kudos [?]: 1 [0], given: 0

Re: GMATprep - exponents & inequalities [#permalink]  20 Aug 2006, 19:58
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
Manager
Joined: 25 Jul 2006
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no wonder I get so many quant questions incorrect due to silly mistakes
Thanks guys!
Senior Manager
Joined: 14 Aug 2006
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Ah.. Thanks Prof.. I was treating these inequalites as if they were true..
Current Student
Joined: 29 Jan 2005
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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)
Manager
Joined: 25 Apr 2006
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Re: GMATprep - exponents & inequalities [#permalink]  24 Aug 2006, 19:18
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.
Manager
Joined: 20 Mar 2006
Posts: 200
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Kudos [?]: 1 [0], given: 0

Re: GMATprep - exponents & inequalities [#permalink]  24 Aug 2006, 21:16
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
Re: GMATprep - exponents & inequalities   [#permalink] 24 Aug 2006, 21:16
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