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

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Skywalker18
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If |x| < x^2, which of the following must be true ? [#permalink] New post   Updated on: 03 Jun 2021, 01:23
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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
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Official Answer
E

Originally posted by Skywalker18 on 19 Jun 2016, 11:13.
Last edited by Bunuel on 03 Jun 2021, 01:23, edited 3 times in total.
Renamed the topic and edited the question.
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adiagr
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Re: If |x| < x^2, which of the following must be true ? [#permalink] New post   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

\(x^2 - 1> 0\)

\(x^2 > 1\)

E is the answer.
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If |x| < x^2, which of the following must be true ? [#permalink] New post   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.

Answer: E.

To understand the underline concept better practice other Trickiest Inequality Questions Type: Confusing Ranges.

Hope it helps.
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Re: If |x| < x^2, which of the following must be true ? [#permalink] New post   21 Jun 2016, 20:39
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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] New post   22 Jun 2016, 08:19
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aditi2013 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
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Re: If |x| < x^2, which of the following must be true ? [#permalink] New post   29 Jun 2016, 23:40
aditi2013 wrote:
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] New post   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] New post   30 Jun 2016, 04:01
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sauravpaul wrote:
aditi2013 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] New post   22 Jul 2016, 22:09
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'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] New post   23 Jul 2016, 00:17
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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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Re: If |x| < x^2, which of the following must be true ? [#permalink] New post   19 Oct 2017, 10:20
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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


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.

Answer: E
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Re: If |x| < x^2, which of the following must be true ? [#permalink] New post   24 Jun 2018, 04:08
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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


It's possible that x=2, since \(|2| < 2^2\).
Eliminate B and D, since they do not have to be true.
It's possible that x=-2, since \(|-2| < (-2)^2\).
Eliminate A and C, since they do not have to be true.

Show Spoiler
E


Algebra:
\(|x| < x^2\)
\(|x| < |x| * |x|\)

Since |x| in the blue inequality must be NONNEGATIVE, we can safely divide each side by |x|:
\(\frac{|x|}{|x|} < |x| * \frac{|x|}{|x|}\)
\(1 < |x|\)

Since each side of the red inequality is NONNEGATIVE, we can safely square each side:
\(1^2 < |x|^2\)
\(1 < x^2\)
\(x^2 > 1\)

Show Spoiler
E
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Re: If |x| < x^2, which of the following must be true ? [#permalink] New post   28 Jun 2018, 09:32
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GMATGuruNY 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


It's possible that x=2, since \(|2| < 2^2\).
Eliminate B and D, since they do not have to be true.
It's possible that x=-2, since \(|-2| < (-2)^2\).
Eliminate A and C, since they do not have to be true.

Show Spoiler
E


Algebra:
\(|x| < x^2\)
\(|x| < |x| * |x|\)

Since |x| in the blue inequality must be NONNEGATIVE, we can safely divide each side by |x|:
\(\frac{|x|}{|x|} < |x| * \frac{|x|}{|x|}\)
\(1 < |x|\)

Since each side of the red inequality is NONNEGATIVE, we can safely square each side:
\(1^2 < |x|^2\)
\(1 < x^2\)
\(x^2 > 1\)

Show Spoiler
E



hey GMATGuruNY great explanation ! :-) i have just one question

Algebra:
\(|x| < x^2\) (as you see \(x^2\) is without bars / brackets )

\(|x| < |x| * |x|\) Here you rewrite \(x^2\)as |x| * |x| and not as x*x , can you please explain why ? :-)

many thanks ! :)
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Re: If |x| < x^2, which of the following must be true ? [#permalink] New post   28 Jun 2018, 10:21
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dave13 wrote:
hey GMATGuruNY great explanation ! :-) i have just one question

Algebra:
\(|x| < x^2\) (as you see \(x^2\) is without bars / brackets )

\(|x| < |x| * |x|\) Here you rewrite \(x^2\)as |x| * |x| and not as x*x , can you please explain why ? :-)

many thanks ! :)


\(x^2 = x * x = |x| * |x|\)

For example:
\(3^2 = 3 * 3 = |3| * |3| = 9\)
\((-3)^2 = -3 * -3 = |-3| * |-3| = 9\)

Given inequality:
\(|x| < x^2\)

Here, I chose to replace x^2 with |x| * |x| so that |x| would appear on both sides, enabling me to divide both sides by |x|:
\(|x| < x^2\)
\(|x| < |x| * |x|\)
\(\frac{|x|}{|x|} < |x| * \frac{|x|}{|x|}\)
\(1 < |x|\)
\(|x| > 1\)
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If |x| < x^2, which of the following must be true ? [#permalink] New post   Updated on: 27 Nov 2023, 23:55
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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


Whenever you are stuck with an equation with an absolute sign, you know you can get rid of the sign by taking positive and negative values of x.

If x >= 0
x < x^2
x^2 - x > 0
x(x - 1) > 0
So x > 1 or x < 0. But x must be positive so x > 1 only.

If x < 0
-x < x^2
x^2 + x > 0
x(x + 1) > 0
So x > 0 or x < -1. But x must be negative so x < -1 only.

We see that x is either > 1 or < -1. This is the same range as option (E).
x^2 - 1 > 0
(x + 1)(x - 1) > 0
x > 1 or x < -1

Answer (E)

Why? Here is an explanation: https://anaprep.com/algebra-the-why-beh ... questions/

Originally posted by KarishmaB on 11 Nov 2019, 22:26.
Last edited by KarishmaB on 27 Nov 2023, 23:55, edited 1 time in total.
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Re: If |x| < x^2, which of the following must be true ? [#permalink] New post   24 Nov 2019, 05:18
@banuel Sir, Can you please quote an example to prove C is not always true?

Thanks in advance
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Re: If |x| < x^2, which of the following must be true ? [#permalink] New post   24 Nov 2019, 05:21
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Sam10smart wrote:
@banuel Sir, Can you please quote an example to prove C is not always true?

Thanks in advance


If x < -1, then C is not true. For example, x = -2 satisfy |x| < x^2 but in this case x > 1 (option C) is not true.
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Re: If |x| < x^2, which of the following must be true ? [#permalink] New post   30 Aug 2020, 08:53
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jamalabdullah100 wrote:
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.

Answer: E.


Hi Bunuel, what am I doing wrong in the below pic?



Bunuel VeritasKarishma Can you please explain whats wrong with this approach. Even I fell for this one!
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If |x| < x^2, which of the following must be true ? [#permalink] New post   Updated on: 27 Nov 2023, 23:56
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Krishchamp wrote:

Bunuel VeritasKarishma Can you please explain whats wrong with this approach. Even I fell for this one!


x(x - 1) > 0
does not imply x > 0 or x > 1

Similarly, x(x+1) > 0
does not imply x > 0 or x > -1

Check this video to know why and to know how to solve these inequalities:
https://youtu.be/PWsUOe77__E

Originally posted by KarishmaB on 02 Sep 2020, 04:49.
Last edited by KarishmaB on 27 Nov 2023, 23:56, edited 1 time in total.
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Re: If |x| < x^2, which of the following must be true ? [#permalink] New post   27 May 2021, 19:28
When x < 0, how can -x < x^2? When you are taking the absolute value of a negative number, isn't the outcome always positive? In this case, when x < 0, shouldn't |-x| = x ? and not -x?

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