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# If x 0, is x^2/|x| < 1?

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Senior Manager
Joined: 27 May 2012
Posts: 455
Re: If x ≠ 0, is x^2/|x| < 1? [#permalink]

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24 Sep 2013, 02:35
Bunuel wrote:
stne wrote:
udaymathapati wrote:
If x#0, is |x|/x<1?

(1) x < 1
(2) x > −1

I think$$\frac{|x|}{x} <1$$

(1) x < 1
(2) x > −1

Here the answer should be E

x= 1/2 satisfies both the statements and answer to the stem is no, 1 is not less 1

X= - 1/2 satisfies both the statements and answer to the stem is yes , -1<1

but for question $$\frac{x^2}{x} <1$$

(1) x < 1
(2) x > −1

here the answer is C as shown above

Please do correct if I am missing something
thanks.

If it were:
If x#0, is |x|/x<1?

(1) x < 1
(2) x > −1

Then the answer is E. The question basically asks whether x is negative and we cannot answer that even when we combine the statements given.

If it were:
If x#0, is x^2/x<1?

(1) x < 1
(2) x > −1

Then the answer is C. The question basically asks whether x<0 or 0<x<1. When we combine the statements, we get that -1<x<1 (x#0). So, the answer to the question is YES.

as you can see the question has now been corrected

originally udaymathapati
had changed x^2 to |x| and posted the question

where is my kudo for pointing this out ?
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- Stne

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Posts: 27
Re: If x 0, is x^2 / |x| < 1? [#permalink]

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10 Oct 2013, 06:48
Bunuel wrote:
zaarathelab wrote:
If x ≠ 0, is x^2 / |x| < 1?
(1) x < 1
(2) x > −1

We can safely reduce LHS of inequality $$\frac{x^2}{|x|}<1$$ by $$|x|$$. We'll get: is $$|x|<1$$? Basically question asks whether $$x$$ is in the range $$-1<x<1$$.

Statements 1 and 2 are not sufficient, but together they are defining the range for $$x$$ as $$-1<x<1$$.

How do you get that $$|x|<1$$?
Math Expert
Joined: 02 Sep 2009
Posts: 44290
Re: If x 0, is x^2 / |x| < 1? [#permalink]

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10 Oct 2013, 07:15
waltiebikkiebal wrote:
Bunuel wrote:
zaarathelab wrote:
If x ≠ 0, is x^2 / |x| < 1?
(1) x < 1
(2) x > −1

We can safely reduce LHS of inequality $$\frac{x^2}{|x|}<1$$ by $$|x|$$. We'll get: is $$|x|<1$$? Basically question asks whether $$x$$ is in the range $$-1<x<1$$.

Statements 1 and 2 are not sufficient, but together they are defining the range for $$x$$ as $$-1<x<1$$.

How do you get that $$|x|<1$$?

$$\frac{x^2}{|x|}<1$$;

$$\frac{|x|*|x|}{|x|}<1$$;

$$|x|<1$$.

Does this make sense?
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Re: If x 0, is x^2/|x| < 1? [#permalink]

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28 Feb 2014, 17:58
Hey Bunuel,
Can you please elaborate how you reduced the left-hand side to |x| < 1? I'm getting confused because of the absolute value sign.
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Joined: 02 Sep 2009
Posts: 44290
Re: If x 0, is x^2/|x| < 1? [#permalink]

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01 Mar 2014, 03:46
2
KUDOS
Expert's post
prsnt11 wrote:
Hey Bunuel,
Can you please elaborate how you reduced the left-hand side to |x| < 1? I'm getting confused because of the absolute value sign.

$$\frac{x^2}{|x|}=\frac{|x|*|x|}{|x|}=|x|$$.

Hope it helps.
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Re: If x 0, is x^2/|x| < 1? [#permalink]

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01 Mar 2014, 12:21
1
KUDOS
Oh yes...so silly of me! If \sqrt{(x^2)} = |x|, x^2 = |x|*|x|
Thanks a lot Bunuel for your prompt help!:)
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Re: If x 0, is x^2/|x| < 1? [#permalink]

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01 Mar 2014, 12:49
1
This post was
BOOKMARKED
msand wrote:
If x ≠ 0, is x^2/|x| < 1?

(1) x < 1
(2) x > -1

Another way i would look at this is : x^2 is always positive, |x| is always positive too .. so a square value divided by the same number would only be less than 1 if the number was a proper fraction instead of an integer .. so the question is asking is x between -1 and 1 or |x| < 1 ?
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Re: If x 0, is x^2 / |x| < 1? [#permalink]

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09 Jun 2015, 23:13
Hi All,

This question is loaded with Number Property rules. If you know the rules, then you can make relatively quick work of this question; you can also solve it by TESTing VALUES...

We're told that X ≠ 0. We're asked if (X^2) /(|X|) < 1. This is a YES/NO question.

Before dealing with the two Facts, I want to point out a couple of Number Properties in the question stem:

1) X^2 will either be 0 or positive.
2) |X| will either be 0 or positive.
3) For the fraction in the question to be LESS than 1, X^2 must be LESS than |X|.

Fact 1: X < 1

IF...
X = 1/2
(1/4)/|1/2| = 1/2 and the answer to the question is YES

IF...
X = -1
(1)/|-1| = 1 and the answer to the question is NO
Fact 1 is INSUFFICIENT

Fact 2: X > −1

IF...
X = 1/2
(1/4)/|1/2| = 1/2 and the answer to the question is YES

IF...
X = 1
(1)/|1| = 1 and the answer to the question is NO
Fact 2 is INSUFFICIENT

Combined, we know...
-1 < X < 1

Since X cannot equal 0, X must be a FRACTION (either negative or positive). In ALL cases, X^2 will be LESS than |X|, so the fraction will ALWAYS be less than 1 and the answer to the question is ALWAYS YES.
Combined, SUFFICIENT

[Reveal] Spoiler:
C

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Re: If x 0, is x^2/|x| < 1? [#permalink]

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13 Mar 2016, 06:05
Here from the question statement
x^2/|x| >1 => the question is actually asking us that does x lies in the (-1,1) range
as you can well see that the combination statement works.
Hence C
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Joined: 08 Dec 2015
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GMAT 1: 600 Q44 V27
Re: If x ≠ 0, is x^2/|x| < 1? [#permalink]

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26 Mar 2016, 11:32
Why can't we manipulate the question to get is x^2<|x| ? can't we pass the |x| to the right? It says in the stem that it's #0. Am i wrong here?
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Re: If x ≠ 0, is x^2/|x| < 1? [#permalink]

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26 Mar 2016, 11:39
iliavko wrote:
Why can't we manipulate the question to get is x^2<|x| ? can't we pass the |x| to the right? It says in the stem that it's #0. Am i wrong here?

We can do this but not because |x| is not 0, but because |x| > 0 and we can multiply both sides by it.
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Re: If x ≠ 0, is x^2/|x| < 1? [#permalink]

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26 Mar 2016, 14:57

Does this mean that the notation x#0 in the question stem doesn't tell anything new? it's already known from the rules that the divisor can't be zero. So if you look at the stem wihtout the x#0 information, the question doesn't change.

Anyways, because whe know that divisor is not zero, we know that x<0 or x>0 and since it's an absolute value, it must be x>0. Is this correct?

Ps. does it mean that in a question where it is mentioned that X#0 but the denominator is not a module, we still must somewhow be sure of the sign of the denominator to be able to multiply like in the case of say, x^2\x so X in both, numerator and denomiator?

Last edited by iliavko on 26 Mar 2016, 15:13, edited 1 time in total.
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Posts: 44290
Re: If x ≠ 0, is x^2/|x| < 1? [#permalink]

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26 Mar 2016, 15:06
iliavko wrote:

Does this mean that the notation x=0 in the question stem doesn't tell anything new? it's already known from the rules that the divisor can't be zero. So if you look at the stem wihtout the x=0 information, the question doesn't change.

Anyways, because whe know that divisor is not zero, we know that x<0 or x>0 and since it's an absolute value, it must be x>0. Is this correct?

Division by 0 is not allowed so $$x \neq 0$$ rules out this case. If we were not told that, then when considering the two statements together we were not be able to tell whether $$\frac{x^2}{|x|}<1$$ because if x= 0 then $$\frac{x^2}{|x|}$$ is undefined not less than 1.

Hope it's clear.
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Re: If x ≠ 0, is x^2/|x| < 1? [#permalink]

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26 Mar 2016, 15:24
Thank you for the replies!

Sorry, but something is still not 100% clear to me, so the meaning of "undefined" is not like "can't possibly be done, so don't even consider it" but more like a valid answer? Isn't undefined a sort of a dead end? Or is it just "correct" to write X#0 with no particular practical reason?
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Re: If x ≠ 0, is x^2/|x| < 1? [#permalink]

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26 Mar 2016, 15:39
iliavko wrote:
Thank you for the replies!

Sorry, but something is still not 100% clear to me, so the meaning of "undefined" is not like "can't possibly be done, so don't even consider it" but more like a valid answer? Isn't undefined a sort of a dead end? Or is it just "correct" to write X#0 with no particular practical reason?

The question asks: is x^2/|x| < 1 if -1 < x < 1. Now, if x is any number but 0 from -1 to 1, then the answer is YES. But if x is 0, then we cannot answer the question, because if x = 0 , then x^2/|x| is undefined.
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Re: If x 0, is x^2/|x| < 1? [#permalink]

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21 Jun 2017, 14:19
Is x^2/|x| < 1?

Which means - is lxl < 1?

or -1 < x < 1?

(1) x < 1 ======> Not Sufficient
(2) x > -1 ======> Not Sufficient

Combining we get: -1 < x < 1

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Re: If x ≠ 0, is x^2/|x| < 1? [#permalink]

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15 Oct 2017, 09:25
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Re: If x ≠ 0, is x^2/|x| < 1?   [#permalink] 15 Oct 2017, 09:25

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