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Intern  Joined: 22 Jun 2010
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If |x| < x^2, which of the following must be true?  [#permalink]

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Question Stats: 52% (01:30) correct 48% (01:41) wrong based on 2276 sessions

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

I. $$x^2 > 1$$

II. $$x > 0$$

III. $$x < -1$$

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only

Originally posted by mehdiov on 20 Aug 2010, 11:59.
Last edited by Bunuel on 09 Apr 2018, 08:58, edited 4 times in total.
Edited the question and added the OA
Math Expert V
Joined: 02 Sep 2009
Posts: 65187
If |x| < x^2, which of the following must be true?  [#permalink]

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18
30
If $$|x| < x^2$$, which of the following must be true?

I. $$x^2 > 1$$

II. $$x > 0$$

III. $$x < -1$$

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only

Given: $$|x|<x^2$$;

Reduce by $$|x|$$: $$1<|x|$$ (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 $$|x|>0$$);

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

I. $$x^2>1$$ --> always true;

II. $$x>0$$ --> may or may not be true;

III. $$x<-1$$ --> may or may not be true.

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Location: Mumbai
Schools: Class of 2008, IIM Ahmedabad
Re: If |x| < x^2, which of the following must be true?  [#permalink]

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7
Hi,

The X-Y plane of looking at it is the fastest way - I would guess.

Mod(x) looks like a V

x^2 is a parabola which just touches the x-axis (tangent) at the origin.

When x < -1; Mod(x) < x^2
When x = -1; Mod (x) = x^2
When -1<x<0; Mod(x) > x^2
When x = 0; Mod (x) = x^2
When 0<x<1; Mod(x) > x^2
When x = 1; Mod (x) = x^2
When x > 1; Mod(x) < x^2

Hope this helps. Thanks.

Wrong forum ?!?!?
##### General Discussion
Intern  Joined: 15 Aug 2010
Posts: 14
Location: Mumbai
Schools: Class of 2008, IIM Ahmedabad
Re: If |x| < x^2, which of the following must be true?  [#permalink]

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2
Bunuel wrote:
III. x<-1 --> --> may or may not be true.

Bunuel, Isn't this always true when |x|<x^2 ... Isn't the answer E.

Thanks.
Math Expert V
Joined: 02 Sep 2009
Posts: 65187
Re: If |x| < x^2, which of the following must be true?  [#permalink]

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4gmatmumbai wrote:
Bunuel wrote:
III. x<-1 --> --> may or may not be true.

Bunuel, Isn't this always true when |x|<x^2 ... Isn't the answer E.

Thanks.

$$|x|<x^2$$ is given as fact and then we asked to determine which of the following statements MUST be true.

$$|x|<x^2$$ means that either $$x<-1$$ or $$x>1$$, $$x$$ can be ANY value from these two ranges, (I think in your own solution you've reached this conclusion: when $$x<-1$$ the graph of $$|x|$$ is below (less than) the graph of $$x^2$$ and when $$x>$$1 again the graph of $$|x|$$ is below the graph of $$x^2$$).

Now, III says $$x<-1$$ this statement is not always true as $$x$$ can be for example 3 and in this case $$x<-1$$ doesn't hold true.

Hope it's clear.
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Manager  Joined: 27 May 2010
Posts: 158
Re: If |x| < x^2, which of the following must be true?  [#permalink]

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I'm confused, because if x= -2 ==> |x|< x^2

|x|< x^2, X^2 should always be > 1
Can eliminate II, because if x=1/2, |x|> x^2

Could you give an example Bunuel
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Joined: 02 Sep 2009
Posts: 65187
Re: If |x| < x^2, which of the following must be true?  [#permalink]

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4
qweert wrote:
I'm confused, because if x= -2 ==> |x|< x^2

|x|< x^2, X^2 should always be > 1
Can eliminate II, because if x=1/2, |x|> x^2

Could you give an example Bunuel

Question is: "which of the following statements MUST be true", not COULD be true.

We are GIVEN that $$|x|<x^2$$, which means that either $$x<-1$$ OR $$x>1$$.

So GIVEN that: $$x<-1$$ OR $$x>1$$.

Statement II. is $$x<-1$$ always true? NO. As $$x$$ could be more than 1, eg. 2, 3, 5.7, ... and in this case $$x<-1$$ is not true. So statement II which says that $$x<-1$$ is not always true.

Hope it's clear.
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Posts: 501
Re: If |x| < x^2, which of the following must be true?  [#permalink]

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mehdiov, what's the OA?

Quote:
Statement II. is x<-1 always true? NO. As x could be more than 1, eg. 2, 3, 5.7, ... and in this case x<-1 is not true. So statement II which says that x<-1 is not always true.

Then we can't even accept I, since x doesn't always have to be greater than 1 (it can be -10, -50 etc.).

The question is similar to (but I still think that it is different):

ps-inequality-13943.html

Maybe I don't understand something.

Thanks.
Math Expert V
Joined: 02 Sep 2009
Posts: 65187
Re: If |x| < x^2, which of the following must be true?  [#permalink]

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1
nonameee wrote:
mehdiov, what's the OA?

Quote:
Statement II. is x<-1 always true? NO. As x could be more than 1, eg. 2, 3, 5.7, ... and in this case x<-1 is not true. So statement II which says that x<-1 is not always true.

Then we can't even accept I, since x doesn't always have to be greater than 1 (it can be -10, -50 etc.).

The question is similar to (but I still think that it is different):

ps-inequality-13943.html

Maybe I don't understand something.

Thanks.

OA is A.

$$|x|<x^2$$ means that either $$x<-1$$ or $$x>1$$. For example $$x$$ could be -5, -3, 2, 3, ...

Basically the question is: If $$x<-1$$ or $$x>1$$ which of the following must be true?

You are saying that answer is: E (I and III only). Is III: $$x<-1$$ ALWAYS true? Is it true for $$x=3$$? NO. So E is not correct.
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Re: If |x| < x^2, which of the following must be true?  [#permalink]

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Hi, If i solve option 1. i get x^2-1>0 or (x+1)(x-1)>0 this implies : x>-1 or x>1.. however the question in the stem can be rephrased as x<-1 or X>1.. How is option 1 true then?am i missing something?

Bunuel wrote:
mehdiov wrote:
If |x|<x^2 , which of the following must be true?

I. x^2>1
II. x>0
III. x<-1

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only

what is the fastest way to solbe it ?

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 $$x>1$$.

I. x^2>1 --> always true;
II. x>0 --> may or may not be true;
III. x<-1 --> --> may or may not be true.

Math Expert V
Joined: 02 Sep 2009
Posts: 65187
Re: If |x| < x^2, which of the following must be true?  [#permalink]

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devinawilliam83 wrote:
Hi, If i solve option 1. i get x^2-1>0 or (x+1)(x-1)>0 this implies : x>-1 or x>1.. however the question in the stem can be rephrased as x<-1 or X>1.. How is option 1 true then?am i missing something?

x^2-1>0 --> x<-1 or x>1.

Solving inequalities:
x2-4x-94661.html#p731476
inequalities-trick-91482.html
data-suff-inequalities-109078.html
range-for-variable-x-in-a-given-inequality-109468.html?hilit=extreme#p873535
everything-is-less-than-zero-108884.html?hilit=extreme#p868863

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

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Bunuel wrote:
mehdiov wrote:
If |x|<x^2 , which of the following must be true?

I. x^2>1
II. x>0
III. x<-1

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only

what is the fastest way to solbe it ?

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 $$x>1$$.

I. x^2>1 --> always true;
II. x>0 --> may or may not be true;
III. x<-1 --> --> may or may not be true.

Bunuel |x|.|x|=x^2 so when we divide x^2 by |x| we get |x|

so |x|.|x|= x^2 is this a formula
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Posts: 65187
Re: If |x| < x^2, which of the following must be true?  [#permalink]

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Joy111 wrote:
Bunuel wrote:
mehdiov wrote:
If |x|<x^2 , which of the following must be true?

I. x^2>1
II. x>0
III. x<-1

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only

what is the fastest way to solbe it ?

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 $$x>1$$.

I. x^2>1 --> always true;
II. x>0 --> may or may not be true;
III. x<-1 --> --> may or may not be true.

Bunuel |x|.|x|=x^2 so when we divide x^2 by |x| we get |x|

so |x|.|x|= x^2 is this a formula

Yes, |x|*|x|=x^2. Consider x=-2 then |x|*|x|=2*2=4=(-2)^2.
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Re: If |x| < x^2, which of the following must be true?  [#permalink]

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Hi Bunuel, I understood your logic that we can safely divide by |x|, however, I tried to solve it this way, just wanted to confirm is this is also right...

x2>|x| => if x is positive then x2-x>0 => x(x-1)>0 => x<0 and x>1................case 1

and if x is negative, then x2+x>0 => x(x+1)>0 =>x<-1 and x>0...case2

From case 1 and case2....x<-1 and x>1..[Same conclusion but lengthier method].. I want to especially confirm because I tried to apply modulus and inequalities theory.. so want to be sure.
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Posts: 65187
Re: If |x| < x^2, which of the following must be true?  [#permalink]

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3
2
pavanpuneet wrote:
Hi Bunuel, I understood your logic that we can safely divide by |x|, however, I tried to solve it this way, just wanted to confirm is this is also right...

x2>|x| => if x is positive then x2-x>0 => x(x-1)>0 => x<0 and x>1................case 1

and if x is negative, then x2+x>0 => x(x+1)>0 =>x<-1 and x>0...case2

From case 1 and case2....x<-1 and x>1..[Same conclusion but lengthier method].. I want to especially confirm because I tried to apply modulus and inequalities theory.. so want to be sure.

x^2>|x|

If x>0 then x2-x>0 --> x(x-1)>0 --> x<0 and x>1. Since we are considering x>0 range then x>1;
If x<0 then x2-x>0 --> x(x+1)>0 --> x<-1 and x>0. Since we are considering x<0 range then x<-1;

So, x^2>|x| holds true for x<-1 and x>1.

Hope it's clear.
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Re: If |x| < x^2, which of the following must be true?  [#permalink]

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1
picked A in 46 secs. for |X| to be less than x^2 just means that it is x is not a decimal. it can be less than -1 and greater than 1. you can only say this in totality with certainty. And square of nay no less than -1 and greater than 1 will be greater than 1. so A
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Re: If |x| < x^2, which of the following must be true?  [#permalink]

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vikram 4689 what x <-1 says is that x can ONLY hold values of less than -1. Which actually means that x cannot have values of 2,3.. etc. this we all know is not true as x can also takes values of >1. Hence this statement cannot hold true. makes sense?
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Re: If |x| < x^2, which of the following must be true?  [#permalink]

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vikram4689 wrote:
Bunuel,
Doesn't the question mean - which of the values of x must hold for |X|<X^2
all values of x<-1 satisfy , although there are other solutions(x>1) but E does not say those solution do not exist. In fact if a superset of values are true then subset of values must be true...

where am mis-interpreting

No, that's not true.

Again: $$|x|<x^2$$ is given as a fact and then we asked to determine which of the following statements MUST be true.

$$|x|<x^2$$ means that $$x<-1$$ or $$x>1$$. Now, evaluate each option:

I. x^2>1. Since $$x<-1$$ or $$x>1$$ then this option is always true;
II. x>0. Since $$x<-1$$ or $$x>1$$ then this option may or may not be true;
III. x<-1. Since $$x<-1$$ or $$x>1$$ then this option may or may not be true.

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

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Hi Bunuel, I tried to solve it this way, please let me know if this is correct as I am getting confused with the final wording of MUST be true. I am kind of stuck at the final stages to make the decision.

The problem mentions: |x|<x2

Case 1: if x is +ve; then x<x2, given that we do not know if x=0, then x-x2<0 ==> x(1-x)<0 ==> multiplying by negative sign; x(x-1)>0...which means x<0 and X>1

Case 2: if x is -ve; then -x<x2, then -x-x2<0 ===>-(x+x2) <0...==> x(1+x)>0 ===> x<-1 and x>0

From C1 AND C2..on the number x<-1 and x>1; after this I am a little lost, how to make the final call.
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Posts: 65187
Re: If |x| < x^2, which of the following must be true?  [#permalink]

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pavanpuneet wrote:
Hi Bunuel, I tried to solve it this way, please let me know if this is correct as I am getting confused with the final wording of MUST be true. I am kind of stuck at the final stages to make the decision.

The problem mentions: |x|<x2

Case 1: if x is +ve; then x<x2, given that we do not know if x=0, then x-x2<0 ==> x(1-x)<0 ==> multiplying by negative sign; x(x-1)>0...which means x<0 and X>1

Case 2: if x is -ve; then -x<x2, then -x-x2<0 ===>-(x+x2) <0...==> x(1+x)>0 ===> x<-1 and x>0

From C1 AND C2..on the number x<-1 and x>1; after this I am a little lost, how to make the final call.

First of all please study these posts:
if-x-x-2-which-of-the-following-must-be-true-99506.html#p767256
if-x-x-2-which-of-the-following-must-be-true-99506.html#p767264
if-x-x-2-which-of-the-following-must-be-true-99506.html#p767437
if-x-x-2-which-of-the-following-must-be-true-99506.html#p776341
if-x-x-2-which-of-the-following-must-be-true-99506.html#p1088622
if-x-x-2-which-of-the-following-must-be-true-99506-20.html#p1089273

Anyway, given that $$|x|<x^2$$, which means that $$x<-1$$ or $$x>1$$. Next we are given three options and are asked which of the following MUST be true.

I. x^2>1. Since $$x<-1$$ or $$x>1$$ then this option is always true (ANY number less than -1 or more than 1 when squared will be more than 1);
II. x>0. This option is NOT always true, for example x could be -2 and in this case x>0 won't not true;
III. x<-1. This option is also NOT always true, for example x could be 2 and in this case x<-1 won't not true.

So, we have that only option I is always true.

Check our Must or Could be True Questions to practice more: search.php?search_id=tag&tag_id=193

Hope it helps.
_________________ Re: If |x| < x^2, which of the following must be true?   [#permalink] 15 Jun 2012, 01:59

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