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Is x > x^2 ? (1) x is greater than x^3 (2) x is greater t

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Is x > x^2 ? (1) x is greater than x^3 (2) x is greater t [#permalink] New post 04 Feb 2012, 08:23
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Is x > x^2 ?

(1) x is greater than x^3

(2) x is greater than x^4


hi can you solve this Agebraically instead of picking numbers
[Reveal] Spoiler: OA

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Last edited by Bunuel on 04 Feb 2012, 08:30, edited 1 time in total.
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Re: x > x^2 [#permalink] New post 04 Feb 2012, 08:29
Expert's post
rxs0005 wrote:
hi can you solve this Agebraically instead of picking numbers

Is x > x^2 ?

S1 x is greater than x^3

S2 x is greater than x^4


Is x > x^2 --> is x(x-1)<0? --> is 0<x<1?

(1) x > x^3 --> x*(x^2-1)<0 --> (x+1)*x*(x-1)<0 --> 0<x<1 or x<-1. Not sufficient.

(2) x > x^4 --> 0<x<1 (only in this range x will be more than x^4). Sufficient.

Answer: B.

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Hope it helps.
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Re: x > x^2 [#permalink] New post 04 Feb 2012, 08:30
thanks B

nice to have u back
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Re: x > x^2 [#permalink] New post 04 Feb 2012, 08:39
B

what I have been realizing ( i feel its wrong now ) is that

x ( x - 1 ) < 0 Means

either x< 0 or x - 1 < 0 or x < 1

so x < 0 or x < 1 how did u get 0 < x < 1 ?
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Re: x > x^2 [#permalink] New post 04 Feb 2012, 08:58
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rxs0005 wrote:
B

what I have been realizing ( i feel its wrong now ) is that

x ( x - 1 ) < 0 Means

either x< 0 or x - 1 < 0 or x < 1

so x < 0 or x < 1 how did u get 0 < x < 1 ?


x(x-1)<0 --> roots are x=0 and x=1 --> "<" sign indicates that the solution lies between the roots: 0<x<1:
Attachment:
1.gif
1.gif [ 891 Bytes | Viewed 1217 times ]


If it were: x(x-1)>0 --> the same roots: x=0 and x=1 --> ">" sign indicates that the solution lies to the left of the smaller root and to the right of the larger root: x<0 or x>1:
Attachment:
2.gif
2.gif [ 997 Bytes | Viewed 1218 times ]


Follow the links in my previous post for more on this subject.

Hope it helps.
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PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: Is x > x^2 ? (1) x is greater than x^3 (2) x is greater t [#permalink] New post 23 Feb 2014, 04:57
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Re: Is x > x^2 ? (1) x is greater than x^3 (2) x is greater t [#permalink] New post 24 Apr 2014, 05:54
Good that you bumped onto this thread Bunuel

Here's my question

On second statement how do we do critical points?

So we have x>x^4

Therefore x^4-x<0

Now then factorizing we have: x^2 (x-1)(x+1) , we only need to take care of (x-1)(x+1) since x^2 is always >=0

But then I get the range -1<x<1 which is not correct given that for instance when x=0 clearly inequality does not hold

Would you mind showing me where I went wrong here?

Thanks!
Cheers
J :)
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Re: Is x > x^2 ? (1) x is greater than x^3 (2) x is greater t [#permalink] New post 24 Apr 2014, 21:31
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jlgdr wrote:
Good that you bumped onto this thread Bunuel

Here's my question

On second statement how do we do critical points?

So we have x>x^4

Therefore x^4-x<0

Now then factorizing we have: x^2 (x-1)(x+1) , we only need to take care of (x-1)(x+1) since x^2 is always >=0

But then I get the range -1<x<1 which is not correct given that for instance when x=0 clearly inequality does not hold

Would you mind showing me where I went wrong here?

Thanks!
Cheers
J :)


Note that the inequality is
x^4 - x < 0

So on factoring you get
x (x^3 - 1) < 0
x(x - 1)(x^2 + x + 1) < 0

Now, (x^2 + x + 1) is always positive. It has no real roots. Hence we might as well assume it to be a positive number and ignore it for the sake of getting our transition points.
Transition points are 0 and 1. Since we want the negative region, the inequality will be negative between 0 and 1.

0 < x < 1

Though, you should try to understand the relation between x, x^2 and x^3 in the ranges < -1, -1 < x < 0, 0 < x <1 and x > 1 so that you don't need to use much algebra for such questions.
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Re: Is x > x^2 ? (1) x is greater than x^3 (2) x is greater t   [#permalink] 24 Apr 2014, 21:31
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