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Is integer x positive? 1. x \gt x^3 2. x \lt x^2

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Is integer x positive? 1. x \gt x^3 2. x \lt x^2 [#permalink]

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New post 06 Aug 2008, 07:29
This topic is locked. If you want to discuss this question please re-post it in the respective forum.

Is integer x positive?

1. \(x \gt x^3\)
2. \(x \lt x^2\)

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Re: Is integer x positive? [#permalink]

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New post 06 Aug 2008, 07:44
A

1) Sufficient. This has to be an integer (fractions would really change up the answer to this one). So with a negative number, when you cube it, you come up with a smaller (to the left on the number line) negative number. x = -2. x^3 = -8. -2 > -8. What if x is positive? x = 2, x^3 = 8. 2 !> 8 ( ! = not in programming). Since we are to take the statements as true, a value that does not conform to the statement is not possible. The only values that are possible to make #1 true are negative numbers. So the answer to the question posed is "No, x is not positive." and the data presented is sufficient.

2) Insufficient. we have x < x^2. x = -2...so x^2 = 4. -2 < 4 => TRUE. x = 2, x = 4 2 < 4 =>TRUE. We have one negative number that works for statement 2 and one positive number for statement 2. This is not enough to give a difinitive answer as to whether x is positive.

aaron22197 wrote:
Is integer x positive?

1. \(x \gt x^3\)
2. \(x \lt x^2\)

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Re: Is integer x positive? [#permalink]

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New post 06 Aug 2008, 08:15
What about x= -1.
-1^1 = -1^3.

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Re: Is integer x positive? [#permalink]

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New post 06 Aug 2008, 08:32
aaron22197 wrote:
Is integer x positive?

1. \(x \gt x^3\)
2. \(x \lt x^2\)



1) x>x^3 (suffcieint)

x^3-x <0 ---> x(x^2-1) <0
we have two solutions
x<0 and x^2-1>0 --(I)
x>0 and x^2-1<0 --(II)
II - this one not the solutions because x is interger.. and x>0 --> x^2-1<0

So we have only one solution i.e x <0 and x^2>1
x can be any -ve integers < -1
i.e -2,-3......................

2) x<x^2
x^2-x>0 -- x(x-1)>0

two solutions
x>0 and x-1>0 (III)
x<0 and x-1<0 (IV)
both are possible solutions (can be +ve or -ve)
Insuffcient

A
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Re: Is integer x positive? [#permalink]

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New post 06 Aug 2008, 08:46
i dont get how A is sufficient ... i rearranged to get x(1-x^2) > 0 ....

this gives me solutions like x>0 or -1<x<1 or x>1 or x<-1 ...

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Re: Is integer x positive? [#permalink]

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New post 06 Aug 2008, 08:51
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pmenon wrote:
i dont get how A is sufficient ... i rearranged to get x(1-x^2) > 0 ....

this gives me solutions like x>0 or -1<x<1 or x>1 or x<-1 ...

x(1-x^2) > 0 for the product tobe >0 both (x and (1-x^2) should have same sign)
two solutions are:
x >0 and (1-x^2)>0 (but this solution is not possible because x is integer and >0 (1-x^2) never >0.
so we can strike out this solution.
or
x <0 and (1-x^2)<0


Got it???
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Re: Is integer x positive? [#permalink]

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New post 06 Aug 2008, 08:52
Great explanation! +1 for you.

x2suresh wrote:
pmenon wrote:
i dont get how A is sufficient ... i rearranged to get x(1-x^2) > 0 ....

this gives me solutions like x>0 or -1<x<1 or x>1 or x<-1 ...

x(1-x^2) > 0 for the product tobe >0 both (x and (1-x^2) should have same sign)
two solutions are:
x >0 and (1-x^2)>0 (but this solution is not possible because x is integer and >0 (1-x^2) never >0.
so we can strike out this solution.
or
x <0 and (1-x^2)<0


Got it???

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Re: Is integer x positive? [#permalink]

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New post 06 Aug 2008, 09:58
sometime plugging in is easy way to solve these kind of problems.

the question says x is an integer. this statement gives half solution.

1: since x is an integer, x cannot be +ve. so it must be -ve. so A is suff.
2: since x is an integer, x can be either -ve or +ve. so not suff.


jallenmorris wrote:
A

1) Sufficient. This has to be an integer (fractions would really change up the answer to this one). So with a negative number, when you cube it, you come up with a smaller (to the left on the number line) negative number. x = -2. x^3 = -8. -2 > -8. What if x is positive? x = 2, x^3 = 8. 2 !> 8 ( ! = not in programming). Since we are to take the statements as true, a value that does not conform to the statement is not possible. The only values that are possible to make #1 true are negative numbers. So the answer to the question posed is "No, x is not positive." and the data presented is sufficient.

2) Insufficient. we have x < x^2. x = -2...so x^2 = 4. -2 < 4 => TRUE. x = 2, x = 4 2 < 4 =>TRUE. We have one negative number that works for statement 2 and one positive number for statement 2. This is not enough to give a difinitive answer as to whether x is positive.

aaron22197 wrote:
Is integer x positive?

1. \(x \gt x^3\)
2. \(x \lt x^2\)

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Re: Is integer x positive? [#permalink]

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New post 06 Aug 2008, 10:18
Pick numbers that satisfy the equation
S1.
x x^3 x>x^3
-1 -1 No (does not satisfy the equation so you can cancel it)
-2 -8 Yes
2 8 NO (does not satisfy the equation so you can cancel it)
For all postivie numbers you will not satisfy S1. So you get all -ve numbers only that satisfy the equation Hence BCE out keep AD

S2.
x x^2 x<x^2
-1 1 Yes
2 4 Yes
You get a Maybe case so D is out

Ans A

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Re: Is integer x positive? [#permalink]

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New post 06 Aug 2008, 15:54
I can't figure out why - but I'm not doing this right. Can tell me why my logic is wrong?

a) x > x^3

Then I simplified to:
0 > x^3 - x
0 > x(x^2-1)
0 > x(x+1)(x-1)

This gives me solution:
0>x, -1>x, 1>x - what this tells me is that x < -1. Ok, I'm good here.

Then I have to test the opposite side of the inequality:
0 < x(x+1)(x-1)

This gives me the solution:
0 < x, x<-1, 1<x - the solution doesn't make sense here. Does it mean it's not valid, therefore the only solution is the one above?

b) x<x^2

I applied the same logic as above, testing both sides of the inequalities.

Both of the solutions were not consistent with each other, so this to me was not sufficient.

Let me know if my logic is flawed. Thanks.

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Re: Is integer x positive? [#permalink]

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New post 06 Aug 2008, 17:09
aaron22197 wrote:
Is integer x positive?

1. \(x \gt x^3\)
2. \(x \lt x^2\)

Consider 4 sets of values of x to solve this :
0<x<1 ----> I
1<x ----->II
x<-1 -------->III
-1<x<0 ------->IV

hence lets solve

(1) x > x^3 => I ,III and IV all values of these sets are satisfied here
hence INSUFFI whether x is positive or negative

(2) x< x^2 => this is satisfied by III,IV and I hence INSUFFI

(1) and (2) :
I,III and IV are satisfying this condition too hence INSUFFI

IMO E :-D
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Re: Is integer x positive? [#permalink]

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New post 06 Aug 2008, 17:12
This thread is really interesting and im waiting for the OA :)

Plsss post the OA
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Re: Is integer x positive? [#permalink]

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New post 10 Aug 2008, 07:39
I think C
1) x>x^3 ->x<1 insuf
2) x<x^2 -> either x>1 or x<0 insuf
Combined -> x<0 suf

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Re: Is integer x positive? [#permalink]

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New post 10 Aug 2008, 09:17
spriya wrote:
aaron22197 wrote:
Is integer x positive?

1. \(x \gt x^3\)
2. \(x \lt x^2\)

Consider 4 sets of values of x to solve this :
0<x<1 ----> I
1<x ----->II
x<-1 -------->III
-1<x<0 ------->IV

hence lets solve

(1) x > x^3 => I ,III and IV all values of these sets are satisfied here
hence INSUFFI whether x is positive or negative

(2) x< x^2 => this is satisfied by III,IV and I hence INSUFFI

(1) and (2) :
I,III and IV are satisfying this condition too hence INSUFFI

IMO E :-D


We are talking about integers here.

Therefore if \(x > x^3\) then x<-1 which is always -ve and is therefore sufficient

Answer is A)

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Re: Is integer x positive?   [#permalink] 10 Aug 2008, 09:17
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