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# Is |x^2|<|x^4|?

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Math Revolution GMAT Instructor
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20 Sep 2017, 23:56
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Is $$|x^2|<|x^4|$$?

1) $$x<-1$$
2) $$|x|<|x^3|$$
[Reveal] Spoiler: OA

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21 Sep 2017, 09:46
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Is $$|x^2|<|x^4|$$ ?

As $$x^2$$ and $$x^4$$ will always be positive , so we have to find if $$x^2 < x^4$$

Also above case will always be true except when -1<x<1
1. as for fractions we know $$(\frac{1}{2})^4 < (\frac{1}{2})^3 <(\frac{1}{2})^2<(\frac{1}{2})$$
that is, for x between -1 and 1 and x => $$x^4 <x^2 <x$$
2. and for x=1 => $$x =x^2 =x^4$$

So here we have to find if -1<x<1 or not.

1) x<−1
Directly tells us equation that we are looking for : that is value of x is not in between -1 and 1. So this makes above statement true.
Sufficient

2) |x|<|x^3|

Here we get magnitude of x is less than magnitude of $$x^3$$. Here magnitude of x cannot be in between 0 and 1
as for x between 0 and 1 $$x^3 < x$$ => $$(1/2)^3 < x$$
And as we are given magnitude comparison we can say x doesn't lie between -1 and 1.
So this makes given question true
Sufficient

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21 Sep 2017, 10:36
MathRevolution wrote:
Is $$|x^2|<|x^4|$$?

1) $$x<-1$$
2) $$|x|<|x^3|$$

Hi Bunuel

I have a doubt here,

Statement 1 specifically mentions that $$x<-1$$ but as per statement 2 either $$x<-1$$ or $$x>1$$. IN either case we will get a definite Yes for the question stem hence answer will be D

But ideally both statements should provide same information.

What am I missing here?

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21 Sep 2017, 10:53
niks18 wrote:
MathRevolution wrote:
Is $$|x^2|<|x^4|$$?

1) $$x<-1$$
2) $$|x|<|x^3|$$

Hi Bunuel

I have a doubt here,

Statement 1 specifically mentions that $$x<-1$$ but as per statement 2 either $$x<-1$$ or $$x>1$$. IN either case we will get a definite Yes for the question stem hence answer will be D

But ideally both statements should provide same information.

What am I missing here?

What contradiction do you see between these two?
_________________

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21 Sep 2017, 10:57
Bunuel wrote:
niks18 wrote:
MathRevolution wrote:
Is $$|x^2|<|x^4|$$?

1) $$x<-1$$
2) $$|x|<|x^3|$$

Hi Bunuel

I have a doubt here,

Statement 1 specifically mentions that $$x<-1$$ but as per statement 2 either $$x<-1$$ or $$x>1$$. IN either case we will get a definite Yes for the question stem hence answer will be D

But ideally both statements should provide same information.

What am I missing here?

What contradiction do you see between these two?

Hi,

My point is that statement 1 says that x<-1 i.e x is negative

But from statement 2 x>1 is also possible and will satisfy the inequality.

Need clarity whether such scenario is possible in a DS question

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21 Sep 2017, 11:02
niks18 wrote:
Bunuel wrote:
niks18 wrote:

Hi Bunuel

I have a doubt here,

Statement 1 specifically mentions that $$x<-1$$ but as per statement 2 either $$x<-1$$ or $$x>1$$. IN either case we will get a definite Yes for the question stem hence answer will be D

But ideally both statements should provide same information.

What am I missing here?

What contradiction do you see between these two?

Hi,

My point is that statement 1 says that x<-1 i.e x is negative

But from statement 2 x>1 is also possible and will satisfy the inequality.

Need clarity whether such scenario is possible in a DS question

On the GMAT, two data sufficiency statements always provide TRUE information and these statements NEVER contradict each other or the stem.

(1) says x < -1.
(2) gives x < -1 or x > 1.

Statements do NOT contradict: together they give x < -1.
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21 Sep 2017, 11:37
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Thanks Bunuel for the clarification

To end the confusion I guess we can directly solve the question as -

Given $$|x^2|<|x^4|$$ or $$\frac{|x^4|}{|x^2|}>1$$

Hence the question stem becomes Is $$|x^2|>1$$

Statement 1: $$x<-1$$, squaring both sides we get $$x^2>1$$ (sign of inequality will reverse because $$|x|>|-1|$$) or

$$|x^2|>1$$. So we get a Yes for our question stem. Hence Sufficient

Statement 2: $$|x|<|x^3|$$ or $$\frac{|x^3|}{|x|}>1$$

Hence $$|x^2|>1$$. Sufficient

Option D

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24 Sep 2017, 18:21
=>

|x^2|<|x^4| ⇔ |x^4| - |x^2| > 0 ⇔ |x^2| ( |x^2| - 1 ) >0
⇔ |x^2| < 0 or |x^2| > 1
⇔ |x| > 1
⇔ x < -1 or x > 1

Condition 1)
x<-1 is sufficient clearly.

Condition 2)
|x|<|x^3| ⇔ |x^3| - |x| > 0 ⇔ |x|( |x^2| - 1 ) > 0
⇔ |x|( |x| + 1 ) ( |x| - 1 ) > 0
⇔ -1 < |x| < 0 or |x| > 1
⇔ |x| > 1
⇔ x < -1 or x > 1
This is sufficient too.

Ans: D
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Re: Is |x^2|<|x^4|?   [#permalink] 24 Sep 2017, 18:21
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