|x + 1|  we set x + 1 = 0 and get x = -1, so

|x + 1| means x + 1 if x > -1 and -x-1 if x < -1  

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|2x - 1|  we set 2x - 1 = 0 and get x = , so

|2x - 1| means 2x - 1 if x >  and -2x +1 if x <  

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|x² + 1| means x² + 1 because x² + 1 is always positive. 

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|1 - 2x²|  We set 1 - 2x² = 0 and get 
                    -2x² = -1
                      x² = 
                       x =  , so

|1 - 2x²| means 1-2x² if  < x <

and -1 + 2x² otherwise.

If this isn't what you mean, you can express what you mean
in the thank you note and I'll email you.

Edwin