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# Is integer x positive?

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Intern
Joined: 27 Sep 2009
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02 Aug 2010, 16:47
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Difficulty:

85% (hard)

Question Stats:

44% (01:37) correct 56% (01:42) wrong based on 199 sessions

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Is integer x positive?

(1) x > x^3
(2) x < x^2
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02 Aug 2010, 17:37
My attempt:

Is integer X positive?

1. X > X^3

A number is greater than number raised to an odd powers implies that the number is an negative integer since a number raised to an odd power would retain its sign.

Ex:

-2 > (-2)^3

Hence X is negative. Sufficient

2. X < X^2

A number raised to an even power would mask its sign. Hence X^2 would always be positive.

Ex :1
-2 < (-2)^2

Ex :2
3 < 3^2

Hence X can be either positive or negative.

Insufficient

Answer is A is alone sufficient to solve this problem.
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Joined: 12 Mar 2010
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03 Aug 2010, 11:53
I'd say C:

The question is: is X>0 ?

1) X > X^3

X can be negative: If X=5, -5 > -75 is true
X can be positive: If X=0,3 0,3 > 0.027 is true

Hence, not sufficient

2) X < X^2

X can be negative: If X=-5, -5<25 is true
X can be positive: If X=2, 2<4 is true

Hence, not sufficient

Together:
The first answer brings two options: either X<0 or 0<X<1
The second answer brings two options: either X<0 or 1<X

X cannot be both greater than 1 and smaller than one, hence it is negative. Answer C
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03 Aug 2010, 12:05
vouzico wrote:
I'd say C:

The question is: is X>0 ?

1) X > X^3

X can be negative: If X=5, -5 > -75 is true
X can be positive: If X=0,3 0,3 > 0.027 is true

Hence, not sufficient

2) X < X^2

X can be negative: If X=-5, -5<25 is true
X can be positive: If X=2, 2<4 is true

Hence, not sufficient

Together:
The first answer brings two options: either X<0 or 0<X<1
The second answer brings two options: either X<0 or 1<X

X cannot be both greater than 1 and smaller than one, hence it is negative. Answer C

I thought a similar thing at first, however on closer inspection I noticed that the original question mentions that X is an integer - therefore we can eliminate the cases where -1<X<1
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03 Aug 2010, 13:19
1
ezhilkumarank wrote:
My attempt:

Is integer X positive?

1. X > X^3

A number is greater than number raised to an odd powers implies that the number is an negative integer since a number raised to an odd power would retain its sign.

Ex:

-2 > (-2)^3

Hence X is negative. Sufficient

2. X < X^2

A number raised to an even power would mask its sign. Hence X^2 would always be positive.

Ex :1
-2 < (-2)^2

Ex :2
3 < 3^2

Hence X can be either positive or negative.

Insufficient

Answer is A is alone sufficient to solve this problem.

1/2 > 1/8

stmt 1 ==> x>-1 and 0<X<1

stmt 2 ==> X<0 and X>1
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03 Aug 2010, 13:26
mehdiov wrote:

1/2 > 1/8

stmt 1 ==> x>-1 and 0<X<1

stmt 2 ==> X<0 and X>1

The questions says that X is an integer. No need to check for x=1/2.

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Re: Is integer X positive? 1. X > X^3 2. X < X^2 I was  [#permalink]

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23 Apr 2013, 10:01
A)$$x > x^3$$ in case 0 < x < 1 or if x < (-1)
but given x is integer so x is -ve number
sufficient
B) $$x < x^2$$
x<0 or
if x>1,
so x could be negative as well as positive. Not sufficient

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

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23 Apr 2013, 11:08
I got it wrong.. i missed the word "integer" in the question.
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Re: Is integer x positive?  [#permalink]

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28 Apr 2013, 03:07
shekar123 wrote:
Is integer x positive?

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

(1) Sufficient. $$x>x^3$$ is the same as $$x^3-x<0$$ or $$x(x^2-1)<0$$.
$$x^2-1>0$$ for any integer $$x$$ except -1, 0 1 which are not appropriate for us. So $$x<0$$

(2) Insufficient. It could be -1 or 2 as well.

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Re: Is integer x positive?  [#permalink]

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19 Jun 2013, 14:38
smyarga wrote:
shekar123 wrote:
Is integer x positive?

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

(1) Sufficient. $$x>x^3$$ is the same as $$x^3-x<0$$ or $$x(x^2-1)<0$$.
$$x^2-1>0$$ for any integer $$x$$ except -1, 0 1 which are not appropriate for us. So $$x<0$$

(2) Insufficient. It could be -1 or 2 as well.

If you put in 1/2 or -2 the result will be the same. but different answers to the question!

Sorry, my mistake oversaw the integer!
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Re: Is integer x positive?  [#permalink]

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19 Jun 2013, 21:06
3
1
shekar123 wrote:
Is integer x positive?

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

You can use inequalities to solve this question or you can also solve it easily if you have a good understanding of the relation between x, x^2 and x^3 in various ranges. In fact, add $$\sqrt{x}$$ to the list as well (though it is not needed in this question).

Method 1: Inequalities

(1) x > x^3
x^3 - x < 0
x(x - 1)(x + 1) < 0
-1, 0 and 1 are the transition points. Inequality will hold when 0 < x <1 or x < -1. So x cannot be integer and positive both. Sufficient.

(2) x < x^2
x^2 - x > 0
x(x - 1) > 0
0 and 1 are the transition points. Inequality will hold when x > 1 or x < 0. x can be an integer and either positive or negative. Not sufficient.

Method 2: It is easy to see that A is enough if you understand the relation between x, x^2 and x^3. Look at the number lines below.

Attachment:

Ques4.jpg [ 9.15 KiB | Viewed 2622 times ]

The green regions show the areas for which the relations hold. You can see that x^3 < x only for negative integer values (and some positive but non integer values) of x. Hence statement 1 is enough.

Check out this link for a similar question that uses understanding of these relations (but the question is much tougher):

http://www.veritasprep.com/blog/2011/08 ... -question/
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Re: Is integer x positive?  [#permalink]

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07 Apr 2015, 02:15
Hi Karishma,

Can you please explain how is x <1 or x > 1
My answer came x > 0 or x > 1.
How to solve that inequality??
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Re: Is integer x positive?  [#permalink]

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07 Apr 2015, 14:45
Hi All,

This question can be approached in a couple of different ways, but you might find that using Number Properties and TESTing VALUES is a pretty fast way to get to the answer.

We're told that X is an INTEGER. We're asked if it is POSITIVE. This is a YES/NO question.

Fact 1: X > X^3

I'm going to TEST VALUES to prove a pattern...

IF....
X = 1
X is NOT > X^3....so X CANNOT be 1

IF...
X = 2
X is NOT > X^3....so X CANNOT be 2

IF....
X=3
X is NOT > X^3....so X CANNOT BE 3

As X increases, X^3 increases even more, so X CANNOT be any positive integer. Since the question asks if X is POSITIVE, we have the answer: NO, X is NEVER positive.
Fact 1 is SUFFICIENT

Fact 2: X < X^2

Here, we can TEST VALUES to prove that there are multiple answers....

IF....
X = 2
X IS < X^2 and the answer to the question is YES.

IF....
X = -1
X is < X^2 and the answer to the question is NO.
Fact 2 is INSUFFICIENT

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Re: Is integer x positive?  [#permalink]

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07 Apr 2015, 20:23
Shree9975 wrote:
Hi Karishma,

Can you please explain how is x <1 or x > 1
My answer came x > 0 or x > 1.
How to solve that inequality??

Check out this post to see how to solve inequalities: http://www.veritasprep.com/blog/2012/06 ... e-factors/

After you read this post, solve x(x - 1)(x + 1) < 0
You will get that inequality will hold when 0 < x <1 or x < -1.
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Re: Is integer x positive?  [#permalink]

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