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# If |x| < x^2, which of the following must be true ?

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If |x| < x^2, which of the following must be true ? [#permalink]

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19 Jun 2016, 11:13
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If |x| < x^2, which of the following must be true ?

A. x > 0
B. x < 0
C. x > 1
D. -1 < x < 1
E. x^2 > 1
[Reveal] Spoiler: OA

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+1 Kudos if you find this post helpful

Last edited by Bunuel on 19 Jun 2016, 11:20, edited 2 times in total.
Renamed the topic and edited the question.
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Re: If |x| < x^2, which of the following must be true ? [#permalink]

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19 Jun 2016, 11:20
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Skywalker18 wrote:
If |x| < x^2, which of the following must be true ?

A. x > 0
B. x < 0
C. x > 1
D. -1 < x < 1
E. x^2 > 1

Given: $$|x|<x^2$$ --> reduce by $$|x|$$ (side note: we can safely do this as absolute value is non-negative and in this case we know it's not zero too, as if x would be zero inequality wouldn't hold true, so $$|x|>0$$) --> $$1<|x|$$ --> $$x<-1$$ or $$x>1$$.

So we have that $$x<-1$$ or $$x>1$$.

A. x > 0. Not always true.
B. x < 0. Not always true.
C. x > 1. Not always true.
D. -1 < x < 1. Not true.
E. x^2 > 1. Always true.

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Re: If |x| < x^2, which of the following must be true ? [#permalink]

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19 Jun 2016, 11:22
Bunuel wrote:
Skywalker18 wrote:
If |x| < x^2, which of the following must be true ?

A. x > 0
B. x < 0
C. x > 1
D. -1 < x < 1
E. x^2 > 1

Given: $$|x|<x^2$$ --> reduce by $$|x|$$ (side note: we can safely do this as absolute value is non-negative and in this case we know it's not zero too, as if x would be zero inequality wouldn't hold true, so $$|x|>0$$) --> $$1<|x|$$ --> $$x<-1$$ or $$x>1$$.

So we have that $$x<-1$$ or $$x>1$$.

A. x > 0. Not always true.
B. x < 0. Not always true.
C. x > 1. Not always true.
D. -1 < x < 1. Not true.
E. x^2 > 1. Always true.

Similar question to practice: if-x-x-2-which-of-the-following-must-be-true-99506.html
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Re: If |x| < x^2, which of the following must be true ? [#permalink]

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21 Jun 2016, 20:39
I used it by plugging in ZONEF.

A. x > 0 : If x is an integer, the inequality holds true. It would be opposite for a fraction
B. x < 0 : Same as A
C. x > 1 : Same as A
D. -1 < x < 1 : Again, we can use Zero to plug in.
E. x^2 > 1 : This must be true.

IMO E.
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Re: If |x| < x^2, which of the following must be true ? [#permalink]

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21 Jun 2016, 22:12
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Skywalker18 wrote:
If |x| < x^2, which of the following must be true ?

A. x > 0
B. x < 0
C. x > 1
D. -1 < x < 1
E. x^2 > 1

|x| < x^2

Both left hand side and Right Hand side are positive, so we can square the inequalities.

$$x^2 < x^4$$

$$x^4 > x^2$$

$$x^4 - x^2 > 0$$
$$x^2(x^2 - 1)> 0$$

x^2 is always positive.

So the condition is

$$x^2 - 1> 0$$

$$x^2 > 1$$

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Re: If |x| < x^2, which of the following must be true ? [#permalink]

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22 Jun 2016, 08:19
I used it by plugging in ZONEF.

A. x > 0 : If x is an integer, the inequality holds true. It would be opposite for a fraction
B. x < 0 : Same as A
C. x > 1 : Same as A
D. -1 < x < 1 : Again, we can use Zero to plug in.
E. x^2 > 1 : This must be true.

IMO E.

your explanation for C is wrong.if u take fraction of x>1 then it satisfies.
But this choice is just sub part of solutions of |x|<x^2 .the other solution is x<-1 which choice C doesn't say about
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Re: If |x| < x^2, which of the following must be true ? [#permalink]

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26 Jun 2016, 01:31
hsbinfy wrote:
I used it by plugging in ZONEF.

A. x > 0 : If x is an integer, the inequality holds true. It would be opposite for a fraction
B. x < 0 : Same as A
C. x > 1 : Same as A
D. -1 < x < 1 : Again, we can use Zero to plug in.
E. x^2 > 1 : This must be true.

IMO E.

your explanation for C is wrong.if u take fraction of x>1 then it satisfies.
But this choice is just sub part of solutions of |x|<x^2 .the other solution is x<-1 which choice C doesn't say about

Aah, thank you for catching this error. The improper fraction will satisfy, right?
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Re: If |x| < x^2, which of the following must be true ? [#permalink]

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29 Jun 2016, 23:40
I used it by plugging in ZONEF.

What is ZONEF? Can you explain this.
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Re: If |x| < x^2, which of the following must be true ? [#permalink]

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29 Jun 2016, 23:43
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I looked for four range of values:

i) x<-1
Is |x| < x^2 in this range? Yes.

ii) -1 < x < 0 N
Is |x| < x^2 in this range? No.

iii) 0 < x < 1 N
Is |x| < x^2 in this range? No.

iv) x > 1 Y
Is |x| < x^2 in this range? Yes.

So, clearly |x| < x^2 only for:

i) x < -1 or
ii) x > 1

In either case, x^2 > 1. So, E.
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Re: If |x| < x^2, which of the following must be true ? [#permalink]

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30 Jun 2016, 04:01
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sauravpaul wrote:
I used it by plugging in ZONEF.

What is ZONEF? Can you explain this.

Hi Saurav,

ZONEF is just an abbreviation to ensure no value remains untested to check the sufficiency. So, I always check ZONEF in majority of such data sufficiency questions.

Zero
One
Negative
Extremes (Lower Limit, High)
Fraction
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Re: If |x| < x^2, which of the following must be true ? [#permalink]

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22 Jul 2016, 22:09
'C' x>1 is a subset of 'E' x^2>1
The question is which of the following must be true so if 'E' must be true then 'C' also must be true?
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Re: If |x| < x^2, which of the following must be true ? [#permalink]

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23 Jul 2016, 00:17
gaurav0480 wrote:
'C' x>1 is a subset of 'E' x^2>1
The question is which of the following must be true so if 'E' must be true then 'C' also must be true?

No. We are given that x < -1 or x > 1. Now, if x < -1, then C is not true.
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If |x| < x^2, which of the following must be true ? [#permalink]

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05 Nov 2016, 23:24
Cant we do this question by opening the Modulus>>??

My approach is as follows
Given that
|x|<x^2

Now opening the modulus accordingly
CASE 1:x<0
=> -x<x^2
=> x+x^2>0
=> x(x+1)>0

By the Line wave Method:
The range of x is x<-1 U x>0
CASE 2: x>0
=> x<x^2
=> x(x-1)>0

Now the range of x is x<0 U x>1

Combining both operations x <-1 U x>1
So C option directly suits or comes in this range..
As per option E.. if I put a value from the above range say x=-0.25..x^2=0.0625 which is less than 1

So according me C should be the correct answer.....

CAN ANYBODY TELL ME WHERE I AM WRONG???
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If |x| < x^2, which of the following must be true ? [#permalink]

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17 Oct 2017, 01:09
Skywalker18 wrote:
If |x| < x^2, which of the following must be true ?

A. x > 0
B. x < 0
C. x > 1
D. -1 < x < 1
E. x^2 > 1

Let's square both the sides, we get,

$$x^4 > x^2$$
->$$x^4 - x^2 > 0$$
-> $$x^2 (x^2 -1) > 0$$
This implies, both the terms have to have same signs.
$$x^2 > 0$$ and $$x^2 - 1 > 0$$.

Additionally, we can't test $$x^2< 0$$ since the square of a real number is never negative.

This gives us $$x^2 > 1$$ (Option E.)
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Re: If |x| < x^2, which of the following must be true ? [#permalink]

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19 Oct 2017, 10:20
Skywalker18 wrote:
If |x| < x^2, which of the following must be true ?

A. x > 0
B. x < 0
C. x > 1
D. -1 < x < 1
E. x^2 > 1

If x = 0, then |0| will not be less than 0^2 since they both are equal to 0. Thus, we know x can’t be 0.

Since |x| = x when x is positive and -x when x is negative, let’s rewrite the inequality without the absolute value sign. That is, if x > 0, then we have x < x^2, and if x < 0, then we have -x < x^2.

Case 1: If x > 0,

x < x^2

Dividing both sides by x, we have:

1 < x

Case 2: If x < 0,

-x < x^2

Dividing both sides by x (switching the inequality sign since x is negative), we have:

-1 > x

Thus, we have x > 1 or x < -1; in that case, x^2 must be greater than 1.

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Re: If |x| < x^2, which of the following must be true ?   [#permalink] 19 Oct 2017, 10:20
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