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Inequality question

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Inequality question

by kaps786 » Wed Aug 17, 2011 6:23 am
Hi Can someone help solve this step by step, shouldnt take long.

Is |x-1| < 1?

(1) (x-1)^2 <= 1
(2) x^2 - 1 > 0



OA follows...

E

Please explain how we evaluate the case when we need to combine statement 1 and statement 2 together i,e evaluating C.
I am a bit unclear on the concept on combing the statements and evaluating then,
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by bblast » Wed Aug 17, 2011 9:25 am
Yup, this is a toughie, I somehow scrambled upto the answer but would be great if someone details the steps involved.
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by Frankenstein » Wed Aug 17, 2011 9:45 am
kaps786 wrote: Is |x-1| < 1?

(1) (x-1)^2 <= 1
(2) x^2 - 1 > 0
Hi,
Basics:
if 'a' is a positive number
|x| < a => -a<x<a
x^2-a^2 < 0 => = -a<x<a
x^2-a^2 > 0 => x< -a or x>a
From(1):
(x-1)^2 - 1^2 <=0
So, -1 <= (x-1) <= 1
This is nothing but |x-1| <=1
|x-1| can be either less than 1 or equal to 1.
Not sufficient

From(2):
x^2-1>0
So, x<-1 or x> 1
So, x-1 < -2 or x-1>0
Not sufficient to say if -1<(x-1)<1

Both(1) and (2):
from(1): -1 <= x-1 <= 1
from(2): x-1 < -2 or x-1 > 0
Combining we get 0 < x-1 <= 1
Still not sufficient to say of -1< (x-1) < 1 because (x-1) = 1 will be a problem.

Hence, E

Checking for different plug-in values is an easier approach. But, just because you are looking for details, I gave this solution.
Cheers!

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by gmatboost » Wed Aug 17, 2011 8:31 pm
I agree that you should remember that if
| ?? | < 1 that means that
-1 < ?? < 1

So in this case we want to know if
-1 < x-1 < 1 ?
I think it is more natural to solve this for x:
0 < x < 2 ?
Maybe on this question it isn't necessary, but in general, when translating the prompt, I think you should get as much clarity about x itself as you can.

Onto the statements:
I think it is easier to think of the inequalities in the statement with the number on the right:

(1) (x-1)^2 <= 1
Think of it for now as ?^2 <= 1

What does it mean if ? squared is less than or equal to 1?
It means that ? itself must be between -1 and 1 (inclusive). Otherwise, the square would be bigger than 1. I like thinking about it this way rather than having to memorize certain formulas, because this way your are encouraged to really analyze what the inequality means.
So, this statement is telling us that -1 <= x-1 <= 1
Or, 0 <= x <= 2
Since x could be 0 or 2, insufficient

Note that if it had said ?^2 <= 4, the solution would be -2 <= ? <= 2
It just happens in this case that 1^2 = 1

Statement 2
x^2 - 1 > 0
Again, I prefer x^2 > 1
This tells me that x must be either below -1 or above 1, otherwise the square would be too small
So, x < -1 or x > 1
x could 1.5 or 3, insufficient

Combined
x < -1 or x > 1
AND
0 <= x <= 2

1 < x <= 2
Is it true that 0 < x < 2?
Since x could be 2, we don't know. Insufficient.
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