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# If x is not equal to 0, is |x| less than 1? (1) x/|x| < x

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Intern
Joined: 06 Oct 2009
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If x is not equal to 0, is |x| less than 1? (1) x/|x| < x [#permalink]

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02 Nov 2009, 04:16
1
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00:00

Difficulty:

85% (hard)

Question Stats:

51% (01:42) correct 49% (01:35) wrong based on 115 sessions

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If x is not equal to 0, is |x| less than 1?

(1) x/|x| < x

(2) |x| > x

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-x-is-not-equal-to-0-is-x-less-than-1-1-x-x-x-86140.html
[Reveal] Spoiler: OA

Last edited by ENGRTOMBA2018 on 17 Nov 2015, 07:47, edited 2 times in total.
TOPIC LOCKED

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Senior Manager
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Re: If x is not equal to 0, is |x| less than 1? (1) x/|x| < x [#permalink]

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02 Nov 2009, 05:34
gmatforce wrote:
If x is not equal to 0, is |x| less than 1?

(1) x/|x| < x

(2) |x| > x

B

(1)
x=-1/2
-1<-1/2

x=2
1<2
Can't say if x is less than 1

(2)
x has to be negative for the absolute value to be greater. Sufficient.

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Re: If x is not equal to 0, is |x| less than 1? (1) x/|x| < x [#permalink]

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02 Nov 2009, 06:25
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I think B is not correct. This question was discussed before, below see my post from the earlier discussion:

x#0, is $$|x|<1$$? Which means is $$-1<x<1$$? (x#0)

(1) $$\frac{x}{|x|}< x$$
Two cases:
A. $$x<0$$ --> $$\frac{x}{-x}<x$$ --> $$-1<x$$. But remember that $$x<0$$, so $$-1<x<0$$

B. $$x>0$$ --> $$\frac{x}{x}<x$$ --> $$1<x$$.

Two ranges $$-1<x<0$$ or $$x>1$$. Which says that x either in the first range or in the second. Not sufficient to answer whether $$-1<x<1$$. (For instance $$x$$ can be $$-0.5$$ or $$3$$)

(2) $$|x| > x$$ Well this basically tells that $$x$$ is negative. But still if we want to see how it works:
Two cases again:
$$x<0$$--> $$-x>x$$--> $$x<0$$.

$$x>0$$ --> $$x>x$$: never correct.

Only one range: $$x<0$$, but still insufficient to say whether $$-1<x<1$$. (For instance $$x$$ can be $$-0.5$$ or $$-10$$)

(1)+(2) $$x<0$$ (from 2) and $$-1<x<0$$ or $$x>1$$ (from 1), hence $$-1<x<0$$. Every $$x$$ from this range is definitely in the range $$-1<x<1$$. Sufficient.

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Re: If x is not equal to 0, is |x| less than 1? (1) x/|x| < x [#permalink]

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02 Nov 2009, 06:34
Bunuel wrote:
I think B is not correct. This question was discussed before, below see my post from the earlier discussion:

x#0, is $$|x|<1$$? Which means is $$-1<x<1$$? (x#0)

(1) $$\frac{x}{|x|}< x$$
Two cases:
A. $$x<0$$ --> $$\frac{x}{-x}<x$$ --> $$-1<x$$. But remember that $$x<0$$, so $$-1<x<0$$

B. $$x>0$$ --> $$\frac{x}{x}<x$$ --> $$1<x$$.

Two ranges $$-1<x<0$$ or $$x>1$$. Which says that x either in the first range or in the second. Not sufficient to answer whether $$-1<x<1$$. (For instance $$x$$ can be $$-0.5$$ or $$3$$)

(2) $$|x| > x$$ Well this basically tells that $$x$$ is negative. But still if we want to see how it works:
Two cases again:
$$x<0$$--> $$-x>x$$--> $$x<0$$.

$$x>0$$ --> $$x>x$$: never correct.

Only one range: $$x<0$$, but still insufficient to say whether $$-1<x<1$$. (For instance $$x$$ can be $$-0.5$$ or $$-10$$)

(1)+(2) $$x<0$$ (from 2) and $$-1<x<0$$ or $$x>1$$ (from 1), hence $$-1<x<0$$. Every $$x$$ from this range is definitely in the range $$-1<x<1$$. Sufficient.

Oh grossly misread the Q as x<1. Thanks for the correction!

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Re: If x is not equal to 0, is |x| less than 1? (1) x/|x| < x [#permalink]

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03 Nov 2009, 20:18
Bunuel, Gmat community's happy to got you.

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Re: If x is not equal to 0, is |x| less than 1? (1) x/|x| < x [#permalink]

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16 Jan 2013, 23:40
1
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gmatforce wrote:
If x is not equal to 0, is |x| less than 1?
(1) x/|x| < x
(2) |x| > x

This is my most feared question type because it requires you to try out values but practice truly reduces that anxiety...

1.
Test x=2: 1 < 2 (This works for the equation but |x| is not less than 1) NO!
Test x=-1/4: -1 < -1/4 (This works for the equation and |x| is less than 1) YES!
INSUFFICIENT.

2. |x| > x
This means x is negative value.
x = -1: |x| is not less than 1 NO!
x = -1/4: |x| is less than 1 YES!
INSUFFICIENT!

Together: We only test (-) values with x/|x| < x
x=-1: No!
x=-1/4: Yes!

So the only valid solution for x/|x| < x that is negative is a fraction.
Fractions are |x| less than 1. YES!

_________________

Impossible is nothing to God.

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Re: If x is not equal to 0, is |x| less than 1? (1) x/|x| < x [#permalink]

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05 Feb 2014, 12:55
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Re: If x is not equal to 0, is |x| less than 1? (1) x/|x| < x [#permalink]

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07 Jun 2015, 13:22
Definitely appears like C to me.

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Re: If x is not equal to 0, is |x| less than 1? (1) x/|x| < x [#permalink]

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07 Jun 2015, 15:19
Hi All,

This DS question is built around some interesting Number Properties and patterns. If you can spot those patterns, then solving this problem should take considerably less time. This also looks like a question that can be beaten by TESTing VALUES.

We're told that X CANNOT = 0. We're asked if |X| < 1. This is a YES/NO question.

Fact 1: X/|X| < X

Before TESTing VALUES, I want to note a pattern in this inequality:

X/|X| will either equal 1 (if X is positive) OR -1 (if X is negative). This will save us some time when it comes to TESTing VALUES, since there are many values of X that will NOT fit this information.

If X = 2, then the answer to the question is NO.

X cannot be 1, any positive fraction, 0, or any negative integer…..

So what's left to TEST….?

If X = -1/2, then the answer to the question is YES.
Fact 1 is INSUFFICIENT

Fact 2: |X| > X

This tells us that X CANNOT be positive or 0.

If X = -1, then the answer to the question is NO.
If X = -1/2, then the answer to the question is YES.
Fact 2 is INSUFFICIENT

Combined, we have deal with the "overlapping restrictions" that we noted in the two Facts:
X cannot be….anything positive, 0, or any negative integer.
X can ONLY BE negative fractions between 0 and -1.
Combined SUFFICIENT.

[Reveal] Spoiler:
C

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Re: If x is not equal to 0, is |x| less than 1? (1) x/|x| < x   [#permalink] 07 Jun 2015, 15:19
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