Question 174215
I think you are making this excessively difficult for yourself.
I can't tell if the given equation was y=|x|- -4 or y-|x|=-4, but I am guessing what you did is mathematically correct, meaning the original equation was y=|x|- -4. So look at that equation compared to the parent function, y=|x|. This is just a "V" with its vertex at (0,0). Because for your equation, each value of y is the parent function's y-value, |x|, minus 4, each point is shifted down 4 units. So this will be a "V" that is shifted down so the vertex is at (0, -4).
Therefore, the domain is all real numbers and the range is all real numbers greater than or equal to -4.