Expert: Ahmed Salami - 2/3/2010

Question
sir can i know the solution of inequality
 ||x|-1|<|1-x|

Answer
Hi Avinash,
We can see that the left and right hand sides are equal when x = 0
||x|-1| = ||0|-1|
       = |0-1|
       = 1
|1-x| = |1-0|
     = 1
Now, we need to examine the intervals x > 0 and x < 0.
When x > 0
||x|-1| = |x-1|
      
|1-x| = |1-x|
     = |x-1|
and so they are equivalent.

When x < 0 (negative, say x = -y where y is positive)
||x|-1| = ||-y| - 1|
       = |y - 1|
      
|1-x| = |1 - -y|
     = |y + 1|
and we can see that ||x|-1| < |1-x|
for all x < 0

Regards