This doesn't follow. If x- 1> 0 and x+ 1> 0 (which happens for x> 1), then |x- 1|= x- 1 and |x+ 1|= x+ 1 so x- 1< x+ 1 from which -1< 1 which is always true.

But if x- 1 and x+ 1 are both negative (which happens for x< -1), then |x- 1|= -(x- 1) and |x+ 1|= -(x+ 1) so -(x- 1)< -(x+ 1) from which x- 1> x+ 1, -1> 1 which is always false.

If -1< x< 1, then x- 1< 0 and x+ 1> 0 so |x- 1|< |x+ 1| becomes -(x- 1)< x+ 1 or -x+ 1< x+ 1 which reduces to -x< x, 0< 2x, x> 0.

The inequality is true for all positive xexceptx= 1. (If the problem had been then it would be true for all positive x.)

Help! I'm clearly doing something wrong here. :/