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# If x^3 < x^2, what is the value of the integer x?

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If x^3 < x^2, what is the value of the integer x?  [#permalink]

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31 Aug 2018, 22:00
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If x^3 < x^2, what is the value of the integer x?

(1) |x| < 2
(2) x^3 = x

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Math Expert
Joined: 02 Aug 2009
Posts: 7108
Re: If x^3 < x^2, what is the value of the integer x?  [#permalink]

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31 Aug 2018, 22:21
1
Bunuel wrote:
If x^3 < x^2, what is the value of the integer x?

(1) |x| < 2
(2) x^3 = x

$$x^3 < x^2.......x^2-x^3>0.....x^2(1-x)>0$$
therefore 1-x>0 or x<1 but $$x\neq{0}$$

(1) |x| < 2
so -2<x<2...
possible values = -1,0,1
but x<1 and not equal to 1, so only value left -1
sufficient

(2) x^3 = x
$$x^3=x.....x^3-x=0......x(x^2-1)=0$$
so x=o or x^2=1 that is -1 and 1
but as seen above only -1 is possible out of -1,0 and 1
sufficient

D
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
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3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Re: If x^3 < x^2, what is the value of the integer x?  [#permalink]

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31 Aug 2018, 22:31
Quote:
If x^3 < x^2, what is the value of the integer x?

(1) |x| < 2
(2) x^3 = x

Please correct me if I am wrong:

Given x^3<x^2
x^3-x^2<0
x^2(x-1)<0
x<0 or x<1
Quote:

STATEMENT 1:

|x|<2:
-2<x<2
means x can be -1,0,1:
If we take x<0 then we will have x=-1 so suff
If we take x<1 then x can be 0 or -1 so two solutions not possible
Hence statement 1 is not sufficient.

Quote:
STATEMENT 2:

x^3=x
x^3-x=0
x(x^2-1)=0
x=0 or |x|=1
we have x= -1,0,1;
Same as statement 1 ; so not sufficient .

combining 1+2

even after clubbing both the statements we have the same result. so the answer must be E.
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If x^3 < x^2, what is the value of the integer x?  [#permalink]

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31 Aug 2018, 22:36
Thanks
chetan2u
I have not counted that x not equal to 0 so the answer i got was completely wrong
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Re: If x^3 < x^2, what is the value of the integer x?  [#permalink]

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31 Aug 2018, 22:39
SonGoku wrote:
Thanks
chetan2u
I have not counted that x not equal to 0 so the answer i got was completely wrong

yes, the question is tricky in that sense, one can easily miss out on this aspect.

therefore look at each question for the hidden meaning too
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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If x^3 < x^2, what is the value of the integer x?  [#permalink]

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31 Aug 2018, 22:40
chetan2u wrote:
Bunuel wrote:
If x^3 < x^2, what is the value of the integer x?

(1) |x| < 2
(2) x^3 = x

$$x^3 < x^2.......x^2-x^3>0.....x^2(1-x)>0$$
therefore 1-x>0 or x<1 but $$x\neq{0}$$

(1) |x| < 2
so -2<x<2...
possible values = -1,0,1
but x<1 and not equal to 1, so only value left -1
sufficient

(2) x^3 = x
$$x^3=x.....x^3-x=0......x(x^2-1)=0$$
so x=o or x^2=1 that is -1 and 1
but as seen above only -1 is possible out of -1,0 and 1
sufficient

D

hi there chetan2u

regarding STATEMENT ONE, i dont get how it can be sufficient.

This is how i understand it

|x| < 2

When X is positive x < 2 so x can be 1 , 0 , -1 etc

now plug in initital question

-1^3 < -1^2, ----> -1<1 TRUE

1^3 < 1^2 ---- > 1<1 NOT TRUE

When x is negative -|x| < 2

-x < 2 (divide by -1)

x > -2 so X can be -1, 0, 1 etc

plug in into initial question

-1^3 < -1^2 ----- > -1 < 1 TRUE

1^3 < 1^2 ----- > 1 < 1 NOT TRUE

So whats wrong with my approach ?

thanks and have a great day
Math Expert
Joined: 02 Aug 2009
Posts: 7108
Re: If x^3 < x^2, what is the value of the integer x?  [#permalink]

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31 Aug 2018, 22:44
1
dave13 wrote:
chetan2u wrote:
Bunuel wrote:
If x^3 < x^2, what is the value of the integer x?

(1) |x| < 2
(2) x^3 = x

$$x^3 < x^2.......x^2-x^3>0.....x^2(1-x)>0$$
therefore 1-x>0 or x<1 but $$x\neq{0}$$

(1) |x| < 2
so -2<x<2...
possible values = -1,0,1
but x<1 and not equal to 1, so only value left -1
sufficient

(2) x^3 = x
$$x^3=x.....x^3-x=0......x(x^2-1)=0$$
so x=o or x^2=1 that is -1 and 1
but as seen above only -1 is possible out of -1,0 and 1
sufficient

D

hi there chetan2u

regarding STATEMENT ONE, i dont get how it can be sufficient.

This is how i understand it

|x| < 2

When X is positive x < 2 so x can be 1 , 0 , -1 etc

now plug in initital question

-1^3 < -1^2, ----> -1<1 TRUE

1^3 < 1^2 ---- > 1<1 NOT TRUE

When x is negative -|x| < 2

-x < 2 (divide by -1)

x > -2 so X can be -1, 0, 1 etc

plug in into initial question

-1^3 < -1^2 ----- > -1 < 1 TRUE

1^3 < 1^2 ----- > 1 < 1 NOT TRUE

So whats wrong with my approach ?

thanks and have a great day

look at the colored portion RED and BLUE
in red you took x as positive but substituted -1 in the equation
and similarly in blue portion, you took x as negative and substituted x as 1

hope you understood
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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If x^3 < x^2, what is the value of the integer x?  [#permalink]

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01 Sep 2018, 19:28
Bunuel wrote:
If x^3 < x^2, what is the value of the integer x?

(1) |x| < 2
(2) $$x^3=x$$

Question stem:- x=?

Given, x is an integer and
$$x^3 < x^2$$
Or, $$x^3-x^2<0$$
Or, $$x^2(x-1)<0$$
Or,$$x:\:\left(-\infty \:,\:0\right)\cup \left(0,\:1\right)$$-------------(a)

St1:- |x| < 2
Or, -2 < x < 2
Or, x: (-2, 2)
Or, x={-1,0,1}------------(b)
From (a)& (b), the only possible integer value of x=-1 (we have an unique value of x)
Sufficient.

St2:- $$x^3=x$$
Or, $$x^3-x=0$$
Or, $$x(x^2-1)=0$$
Or, $$x(x+1)(x-1)=0$$
x={-1,0,1}-----------------------(c)

From (a)& (c), the only possible integer value of x=-1 (we have an unique value of x)
Sufficient

Ans. (D)
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Re: If x^3 < x^2, what is the value of the integer x?  [#permalink]

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01 Sep 2018, 21:35
1
Bunuel wrote:
If x^3 < x^2, what is the value of the integer x?

(1) |x| < 2
(2) x^3 = x

chetan2u

i have a doubt

in the question stem ---> x^3 - x^2 < 0;
then x^2 - x^3 > 0
x^2(1 - x^2) > 0

so since the product is greater than zero, therefore x can never be Zero (right?)
secondly,
x^2 > 0;
x > 0

and

1 > x^2
|x| < 1 or -1 < x < 1

is this the correct way to break the stem?
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Joined: 26 Mar 2013
Posts: 1919
Re: If x^3 < x^2, what is the value of the integer x?  [#permalink]

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02 Sep 2018, 04:44
Bunuel wrote:
If x^3 < x^2, what is the value of the integer x?

(1) |x| < 2
(2) x^3 = x

x^3 < x^2 means 2 things:

1) x does not equal to 0
2) x^2 is always positive,

Therefore, we can divide both sides safely by x^2 WITHOUT flipping the sign.

It will be if x < 1, Integer x value?

(1) |x| < 2

-2< |x| < 2.........3 intgere values 1, 0 ,-1

0 & 1 are invalid values.........So the only integers left is -1........We have unique value

Sufficient

(2) x^3 = x

Refer to note 1 above( zero is not considered), we can divide by x...Hence

x^2 =1........Only value is -1 (It can't be 1 as stem says x<1)

Sufficient

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Posts: 1919
If x^3 < x^2, what is the value of the integer x?  [#permalink]

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Updated on: 02 Sep 2018, 04:57
saurabh9gupta wrote:
Bunuel wrote:
If x^3 < x^2, what is the value of the integer x?

(1) |x| < 2
(2) x^3 = x

chetan2u

i have a doubt

in the question stem ---> x^3 - x^2 < 0;
then x^2 - x^3 > 0
x^2(1 - x^2) > 0

so since the product is greater than zero, therefore x can never be Zero (right?)
secondly,
x^2 > 0;
x > 0

and

1 > x^2
|x| < 1 or -1 < x < 1

is this the correct way to break the stem?

saurabh9gupta

The part in redis incorrect.

x^2(1 - x) > 0...It should be x, after factoring out x^2.

Also the highlighted part is incorrect

x^2 >0 is ok but it is incorrect to conclude that only x > 0

2 > 0 ......2^2 >0

-2 <0 ......but also -2^2 >0

So x could take negative or positive value

If I continue your way above, I would do the following:

x^2 is +ve and does not equal to zero...So divide both sides safely ........So

1- x <.....1<x

I hope it helps

Originally posted by Mo2men on 02 Sep 2018, 04:49.
Last edited by Mo2men on 02 Sep 2018, 04:57, edited 1 time in total.
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Re: If x^3 < x^2, what is the value of the integer x?  [#permalink]

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02 Sep 2018, 04:56
Bunuel wrote:
If x^3 < x^2, what is the value of the integer x?

(1) |x| < 2
(2) x^3 = x

Statemnt I:
Given, $$x^2(1-x) > 0$$.. So, x can be -1,-2,-3.. etc.
But as $$|x| <2$$, x can be only -1.............Sufficient.

Statement II:
From, this statement - $$x = 0, x = -1$$
But x can't be 0 as $$x^2(1-x) > 0$$. Hence, x = -1

Sufficient.
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Re: If x^3 < x^2, what is the value of the integer x? &nbs [#permalink] 02 Sep 2018, 04:56
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