abs [x]= -x not abs -[x] = -x any formula, restrictions, or imaginary numbers to produce the negative x?

Question

on the first formula? Thinking along the lines of abs [-x] < 0 or something similar.

Answer 1

Well, any time x < 0, abs(x) = -x. E.g., if x = -2, then abs(-2) = 2 = -(-2).

If you're asking if there are any situations where abs(x) < 0, the answer is no; by convention, any function denoted as |∙| (abs) maps onto the non-negative real numbers. This is true for complex (imaginary) numbers as well as reals, as well as vectors, as well as ... well, any metric space. [Sometimes ||∙|| is used instead of |∙|, but they basically mean the same thing.]
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Answer 2

I'm not quite sure what you want. |x| = -x iff x is a nonpositive real number. Note that |-x| = |x|, so again, |-x| = -x iff x is a nonpositive real number.

Note that even though you can have |x| = -x, this will only happen when x is nonpositive -- i.e. when -x is nonnegative. You will never have |x| be a negative number -- indeed, this is the essential characteristic of the absolute value function, namely that it always outputs a nonnegative real number.