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# X / |X| < x What must be true about X? x > 1 x > -1

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CEO
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X / |X| < x What must be true about X? x > 1 x > -1 [#permalink]  01 Nov 2007, 11:58
X / |X| < x

What must be true about X?

x > 1
x > -1
|X| < 1
|X| = 1
|X|^2 > 1
Manager
Joined: 26 Sep 2007
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Kudos [?]: 11 [0], given: 5

x >1

If X =1, 1/1 < 1 which is not true
If X -ve, let's say -1, then we get -1/1 < -1
-1 < -1 which is not true
CEO
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Re: Absolute Expressions [#permalink]  01 Nov 2007, 13:02
bmwhype2 wrote:
X / |X| <x> 1
x > -1
|X| <1> 1

X/X<X or X/-X<X

So 1<X or -1<X Essentially X is greater than both 1 and -1.

A: Looks good, satifies both
B: satifies only one condition. Out
C: This says that -1<x<1>1 or -(x^2)>1,

x>1 or x<-1 and I don't think we can sqrt the second one -(x^2)>1 b/c this would make an imaginary number. plz confirm this someone. Thx.

Anyway. I say A.
Director
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Re: Absolute Expressions [#permalink]  01 Nov 2007, 13:14
bmwhype2 wrote:
X / |X| <x> 1
x > -1
|X| <1> 1

i say B - FIG PLEASE CONFIRM!!

for positive :
x/x <x
1 < x or x > 1

for negative :
-x/x > x
-1 > x or -1 < x < 0

A doesn't necessarily have to be true. x could be greater than 1 or could be between -1 and 0 in the negative

B satisfies both the negative and positive conditions for x

C not possible

D is impossible

E not necessarily true
Senior Manager
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Re: Absolute Expressions [#permalink]  01 Nov 2007, 13:31
bmwhype2 wrote:
X / |X| <x> 1
x > -1
|X| <1> 1

A is correct.

(A) x > 1
Assume x = 2 then we get 2 / |2| = 1 which is LESS than 2. Hence, correct.
This would be true for any value of x > 1

(B) x > -1
Assume x = 0.5 then we get 0.5 / |0.5| = 1 which is MORE than 0.5. Hence, wrong.
So all the values of x > -1 do not satisfy X / |X| < x
SVP
Joined: 01 May 2006
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(B)... I confirm

X / |X| < x
<=> X / |X| - x < 0
<=> x*(1-|x|) < 0 as |x| > 0 and this inequation is not using x = 0 as a possibility, which is prohibited
<=> x*(|x| - 1) > 0
<=> sign(x) = sign(|x|-1)

In other word, we have to know
o for which x > 0, |x|-1 > 0
o for which x < 0, |x|-1 < 0

o If x > 0, then
|x|-1 > 0
<=> x - 1 > 0
<=> x > 1

o If x < 0, then
|x|-1 < 0
<=> -x - 1 < 0
<=> x > -1.... So 0 > x > -1

That means :
0 > x > -1 OR x > 1.... In other word x must be superior to -1.
SVP
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Re: Absolute Expressions [#permalink]  01 Nov 2007, 13:45
gluon wrote:
bmwhype2 wrote:
X / |X| <x> 1
x > -1
|X| <1> 1

A is correct.

(A) x > 1
Assume x = 2 then we get 2 / |2| = 1 which is LESS than 2. Hence, correct.
This would be true for any value of x > 1

(B) x > -1
Assume x = 0.5 then we get 0.5 / |0.5| = 1 which is MORE than 0.5. Hence, wrong.
So all the values of x > -1 do not satisfy X / |X| < x

There is a trick in the question... It states "must be true".... x must be superior to -1... but we do not say x=0.5...
CEO
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Re: Absolute Expressions [#permalink]  01 Nov 2007, 14:24
Fig wrote:
gluon wrote:
bmwhype2 wrote:
X / |X| <x> 1
x > -1
|X| <1> 1

A is correct.

(A) x > 1
Assume x = 2 then we get 2 / |2| = 1 which is LESS than 2. Hence, correct.
This would be true for any value of x > 1

(B) x > -1
Assume x = 0.5 then we get 0.5 / |0.5| = 1 which is MORE than 0.5. Hence, wrong.
So all the values of x > -1 do not satisfy X / |X| <x>-1 satisfies all the values.

B is correct.
Senior Manager
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Re: Absolute Expressions [#permalink]  01 Nov 2007, 16:32
I am not understanding the trick here. The question asks if x > -1 when it is given that X / |X| < x

0.5 > -1 and we can see that x = 0.5 does NOT satisfy X / |X| <x

We can also verify that all values x > 1 satisfy X / |X| < x

So how is B correct?

Fig wrote:
There is a trick in the question... It states "must be true".... x must be superior to -1... but we do not say x=0.5...
Intern
Joined: 21 Jun 2007
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B is not saying everything greater than (-1) WORKS. It just states every value of x that DOES work must be > (-1). In order to kill B you must find something smaller than (-1) that works, and that seems impossible.

(-.5) seems to kill choice A, no? It's smaller than 1 and it gives us (-1) < (-.5). No problem there. So I'm going with B.
Senior Manager
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*Sigh* never mind... I dont understand this question. My mind does not comprehend such twisted logic.
Manager
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Ok, plugging any value > -1 makes it correct, it's B.

Fig or beckee529, help me understand this. Correct me what I am doing wrong below:

For +ve X,

X/X < X yields 1 <X> 1

For -ve X,

-X/-X < -X yields 1 < -X or we can say X < -1
When I say X < -1, it means X can be -2, -3, -4.........etc.

These absolute questions are tricky.

Last edited by sevenplus on 02 Nov 2007, 11:43, edited 1 time in total.
Intern
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Plug in the 'usual values' : -1, 0, 1, between 0 and -1, less than -1, between 0 and 1, greater than 1....
you'll find that

X must be greater than 1 (X>1) and that X must be between -1 and 0
(-1<X<0)...
SVP
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sevenplus wrote:
Ok, plugging any value > -1 makes it correct, it's B.

Fig or beckee529, help me understand this. Correct me what I am doing wrong below:

For +ve X,

X/X < X yields 1 <X> 1

For -ve X,

-X/-X < -X yields 1 < -X or we can say X < -1
When I say X < -1, it means X can be -2, -3, -4.........etc.

These absolute questions are tricky.

It's in your way to replace |x|/x < x when x is negative.... U have to replace only |x| by -x. The rest remains unchanged as they do not have to be positive all time
Manager
Joined: 26 Sep 2007
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Fig wrote:
sevenplus wrote:
Ok, plugging any value > -1 makes it correct, it's B.

Fig or beckee529, help me understand this. Correct me what I am doing wrong below:

For +ve X,

X/X < X yields 1 <X> 1

For -ve X,

-X/-X < -X yields 1 < -X or we can say X < -1
When I say X < -1, it means X can be -2, -3, -4.........etc.

These absolute questions are tricky.

It's in your way to replace |x|/x < x when x is negative.... U have to replace only |x| by -x. The rest remains unchanged as they do not have to be positive all time

Fig, thanks for pointing out the wrong step. It makes sense now.
Current Student
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Re: Absolute Expressions [#permalink]  03 Nov 2007, 14:44
Okay, I'm not clear on the output of calculating the negative part of the equation: X/|X|<X = -1<X<0.

How do you get X<0? I understand how you're getting X>-1, but don't see how you're getting X<0. Please clarify, thanks!
Director
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Re: Absolute Expressions [#permalink]  03 Nov 2007, 14:50
hd54321 wrote:
Okay, I'm not clear on the output of calculating the negative part of the equation: X/|X|<X = -1<X<0.

How do you get X<0? I understand how you're getting X>-1, but don't see how you're getting X<0. Please clarify, thanks!

since we are solving for the negative part the range can only be less than zero.. so the range found here is between -1 and 0
CEO
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Assuming that x can't be 0, x/|x| is either 1 or -1, so...

-1 < x
OR
1 < x

This means that x HAS to be greater than -1, since greater than 1 is always greater than -1.
Manager
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--corrected

Late, but B:

+ve:
x/x < x
1 < x --> x > 1

-ve:
-x/x < x
-1 < x --> x > -1

So X > -1

Last edited by Whatever on 17 Dec 2007, 22:30, edited 1 time in total.
CEO
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Expert's post
B

X / |X| < X, X e (-1,0)&(1,+∞)

Therefor X>-1 cover all X satisfied the inequality.

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