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Is x between 0 and 1?

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Director
Joined: 07 Jun 2004
Posts: 610

Kudos [?]: 949 [0], given: 22

Location: PA
Is x between 0 and 1? [#permalink]

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12 Feb 2012, 15:07
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Difficulty:

55% (hard)

Question Stats:

48% (00:28) correct 52% (01:30) wrong based on 25 sessions

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Is x between 0 and 1

(1) x^2 > x^3
(2) -x > x^3

[Reveal] Spoiler:
I solved this as below

x^3 - x^2 < 0 or x^2( x - 1 ) < 0 roots will be x = 0 and x = 1 or

0 < x < 1 why is this wrong ?
[Reveal] Spoiler: OA

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If the Q jogged your mind do Kudos me : )

Kudos [?]: 949 [0], given: 22

Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4488

Kudos [?]: 8751 [2], given: 105

Re: Is x between 0 and 1 [#permalink]

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12 Feb 2012, 15:46
2
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Expert's post
Hi, there. I'm happy to help with this.

Prompt:
Is x between 0 and 1?

Statement #1: x^2 > x^3

Consider four categories of numbers
(a) positive numbers bigger than one
(b) positive numbers between one and zero
(c) negative numbers between 0 and -1
(d) negative numbers less than -1 (i.e. with absolute value greater than 1)

This is always a good list to keep in mind, especially on DS questions on exponents.

Consider the four categories:
(a) x = 2 ---> 4 < 8, statement #1 is not true
(b) x = 0.5 ---> 0.25 > 0.125, statement #1 is true
(c) x = -0.5 ---> +0.25 > -0.125, statement #1 is true
(d) x = -2 ---> +4 > -8, statement #1 is true

So, statement #1 allows for several categories, not just between 0 and -1. Statement #1 is insufficient.

Statement #2: -x > x^3

If x is positive, this will never be true, because -x would be negative, x^3 would be positive, and any positive is greater than any negative.

If x is negative, this will always be true, because -x would be positive, x^3 would be negative, and any positive is greater than any negative.

If x is negative, it's not between 0 and 1. This allows us to give a conclusive answer to the prompt. Statement #2, by itself, is sufficient.

The problem with your algebraic method -- you factored correctly, and arrived at:

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

but this is a cubic inequality. While you do need to find the roots, it's not enough just to say that the solution exists between them. The roots delimit regions of the number line, and we need to test the inequality in each region. Thus,

Region I = less than 0
Region II = between 0 and 1
Region III = greater than 1

Upon testing values, we find that numbers in both Region I and Region II satisfy this inequality. Thus, this inequality by itself does not guarantee that the numbers are or are not between 0 and 1.

Does that make sense? Please let me know if you have questions on anything I've said here.

Mike
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Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)

Kudos [?]: 8751 [2], given: 105

Math Expert
Joined: 02 Sep 2009
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Re: Is x between 0 and 1 [#permalink]

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13 Feb 2012, 06:40
rxs0005 wrote:
Is x between 0 and 1

1. x^2 > x^3

2. -x > x^3

I solved this as below

x^3 - x^2 < 0 or x^2( x - 1 ) < 0 roots will be x = 0 and x = 1 or

0 < x < 1 why is this wrong ?

the OA is B

Is x between 0 and 1

(1) x^2 > x^3 --> x^2*(x-1)<0, since x^2 here must be positive, then x-1 must be negative: x-1<0 --> x<1. Not sufficient.

(2) -x>x^3 --> x(x^2+1)<0 --> since x^2+1 is positive then x must be negative: x<0, so the answer to the question is NO. Sufficient.

Hope it's clear.
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Director
Status: Finally Done. Admitted in Kellogg for 2015 intake
Joined: 25 Jun 2011
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Location: United Kingdom
GMAT 1: 730 Q49 V45
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is x between 0 & 1 [#permalink]

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01 Apr 2012, 12:34
Is x between 0 and 1?

(1) $$x^2 > x^3$$
(2) $$-x > x^3$$

In these questions is it worth factoring out the inequality or just pick the numbers? And also what are the best numbers to pick?
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E.

MGMAT 1 --> 530
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Kudos [?]: 4216 [0], given: 217

Math Expert
Joined: 02 Sep 2009
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Kudos [?]: 132781 [0], given: 12372

Re: is x between 0 & 1 [#permalink]

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01 Apr 2012, 12:38
enigma123 wrote:
Is x between 0 and 1?

(1) $$x^2 > x^3$$
(2) $$-x > x^3$$

In these questions is it worth factoring out the inequality or just pick the numbers? And also what are the best numbers to pick?

Merging similar topics. There are both number picking and algebraic approaches shown above. Please ask if anything remains unclear.
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Kudos [?]: 132781 [0], given: 12372

Re: is x between 0 & 1   [#permalink] 01 Apr 2012, 12:38
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