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

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Director
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If x is not equal to 0, is |x| less than 1? [#permalink]  06 Jan 2010, 12:31
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If x is not equal to 0, is |x| less than 1?

(1) x/|x| < x
(2) |x| > x
[Reveal] Spoiler: OA
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Re: |x| less than 1? [#permalink]  06 Jan 2010, 13:00
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ugimba wrote:
If x is not equal to 0, is |x| less than 1?

(1) $$x/|x| < x$$
(2) |x| > x

IMO A...

ST 1: can be written as $$x < x * |x|$$ since multiplying by |x| which is always positive would not change the sign.

This gives x as -1<x<0.... therefore sufficient...

ST2: |x| > x gives value of x as x<0... hence could be -2,-3.... Therefore not sufficient...

Cheers!
JT
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Re: |x| less than 1? [#permalink]  06 Jan 2010, 14:27

X<X|x| ==> 1<|x| if X >0
==>1>|x| if X<0
so to conclude we need to know id X<0 or X>0

whats the OA??
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Re: |x| less than 1? [#permalink]  06 Jan 2010, 14:29
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jeeteshsingh wrote:
ugimba wrote:
If x is not equal to 0, is |x| less than 1?

(1) $$x/|x| < x$$
(2) |x| > x

IMO A...

ST 1: can be written as $$x < x * |x|$$ since multiplying by |x| which is always positive would not change the sign.

This gives x as -1<x<0.... therefore sufficient...

ST2: |x| > x gives value of x as x<0... hence could be -2,-3.... Therefore not sufficient...

Cheers!
JT

1) this only holds true if x is a negative fraction or greater than 1. The question however is asking if |x| is less than 1. The abs val of a number less than -1 will be greater than 1, whereas the abs val of a negative fraction will be less than 1.

x/|x| < x

try -2, -2/|-2| = -1, -1 is greater than -2, so does not hold
try -1/2, (-1/2)/|(-1/2)| = -1, -1 is less than -1/2 so this holds.
try 1/2, (1/2)/|(1/2)| = 1, 1 is greater than 1/2, so this does not hold
try 2, 2/|2| = 1, 1 is less than 2, so this holds

so, -1 < x < 0 and x > 1

the question asks: is |x| < 1?

-1 < x < 0 --> |x| will be a positive fraction, i.e. less than 1
x > 1 --> |x| will be greater than 1

hence insufficient.

2) the abs val of a positive number equals that number so this only holds true for negative numbers (including fractions). The question is asking if |x| is less than 1.
Same as in (1),
|x| > x

x < 0

try a negative integer, -2,
| -2 | = 2, --> |x| will be greater 1

try a negative fraction, -(1/2),
| -(1/2) | = 1/2 --> |x| will be less than 1

hence insufficient.

putting both together,

x < 0 and -1< x < 0 and x > 1
this limits x to -1< x < 0, thus |x| will be a positive fraction and will always be less than 1.

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Re: |x| less than 1? [#permalink]  06 Jan 2010, 14:44
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firasath wrote:
jeeteshsingh wrote:
ugimba wrote:
If x is not equal to 0, is |x| less than 1?

(1) $$x/|x| < x$$
(2) |x| > x

IMO A...

ST 1: can be written as $$x < x * |x|$$ since multiplying by |x| which is always positive would not change the sign.

This gives x as -1<x<0.... therefore sufficient...

ST2: |x| > x gives value of x as x<0... hence could be -2,-3.... Therefore not sufficient...

Cheers!
JT

1) this only holds true if x is a negative fraction or greater than 1. The question however is asking if |x| is less than 1. The abs val of a number less than -1 will be greater than 1, whereas the abs val of a negative fraction will be less than 1.

x/|x| < x

try -2, -2/|-2| = -1, -1 is greater than -2, so does not hold
try -1/2, (-1/2)/|(-1/2)| = -1, -1 is less than -1/2 so this holds.
try 1/2, (1/2)/|(1/2)| = 1, 1 is greater than 1/2, so this does not hold
try 2, 2/|2| = 1, 1 is less than 2, so this holds

so, -1 < x < 0 and x > 1

the question asks: is |x| < 1?

-1 < x < 0 --> |x| will be a positive fraction, i.e. less than 1
x > 1 --> |x| will be greater than 1

hence insufficient.

2) the abs val of a positive number equals that number so this only holds true for negative numbers (including fractions). The question is asking if |x| is less than 1.
Same as in (1),
|x| > x

x < 0

try a negative integer, -2,
| -2 | = 2, --> |x| will be greater 1

try a negative fraction, -(1/2),
| -(1/2) | = 1/2 --> |x| will be less than 1

hence insufficient.

putting both together,

x < 0 and -1< x < 0 and x > 1
this limits x to -1< x < 0, thus |x| will be a positive fraction and will always be less than 1.

I second that... my slip.... i missed the greater part in ST 1...... ! Yes.. it should be C....
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Re: |x| less than 1? [#permalink]  06 Jan 2010, 14:49
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firasath, fantastic explanation. I missed taking fractions into account.

+1 for ya.
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Re: |x| less than 1? [#permalink]  06 Jan 2010, 14:54
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Expert's post
jeeteshsingh wrote:
ugimba wrote:
If x is not equal to 0, is |x| less than 1?

(1) $$x/|x| < x$$
(2) |x| > x

IMO A...

ST 1: can be written as $$x < x * |x|$$ since multiplying by |x| which is always positive would not change the sign.

This gives x as -1<x<0.... therefore sufficient...

ST2: |x| > x gives value of x as x<0... hence could be -2,-3.... Therefore not sufficient...

Cheers!
JT

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

Q: is $$-1<x<1$$ true?

(1) $$\frac{x}{|x|} < x$$:

$$x<0$$ --> $$\frac{x}{-x} < x$$ --> $$x>-1$$, but as $$x<0$$, then --> $$-1<x<0$$;

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

So x can be as in the range {-1,1} as well as out of this range. Not sufficient.

(2) $$|x| > x$$ --> $$x<0$$. Not sufficient.

(1)+(2) Intersection of the ranges from (1) and (2) is: $$-1<x<0$$. Every $$x$$ from this range is in the range {-1,1}. Sufficient.
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Re: |x| less than 1? [#permalink]  07 Jan 2010, 10:54

1. For Condition 1, x can either be positive integer or -ve fraction. Its true for both i.e if x = 2,3, etc
or x = -1/2, -1/3, etc.
Hence not sufficient.

2. For condition 2, X has to be -ve fraction or integer. Its true only when x = -1/2, -1/3, etc. or x=-1, -2, etc.
Hence B not sufficient.

3.Combine 1 & 2. True only when x is -ve fraction i.e. x=-1/2, -1/3, etc.
Hence |x| < 1.
Hence C should be the answer.
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Re: |x| less than 1? [#permalink]  10 Jan 2010, 09:50
mod(x)/(x) can be 1 or -1
x ranges from (-1) to (infinity)
or (1) to infinity.

mod(x) > x means x is definitely a negative number

so x is between - 1 and 0 and is hence a fraction.
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Re: If x is not equal to 0, is |x| less than 1? [#permalink]  06 Feb 2014, 02:32
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Re: If x is not equal to 0, is |x| less than 1? [#permalink]  03 Oct 2015, 10:35
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