Show that for every x in R , 0<=|x|-x<=2|x|

hello
help me please!

To show that |x| - x >= 0, we just note that |x| >= x always.

To show that |x|-x <= 2|x|, we can break it into 2 cases:

Case 1: x >= 0. Then |x| = x, and so:

|x|-x = x - x = 0.

and clearly 0 <= 2|x|, so we are done.

Case 2: x < 0. Then |x| = -x, and so:

|x|-x = -x -x = 2x <= 2|x|

Done.