For the function to be defined, the denominator |x| - x must not be zero. |x| - x = 0 occurs when |x| = x, which is true for all x ≥ 0. Therefore, x cannot be zero or any positive number. However, for x < 0, |x| = -x, so the denominator becomes -x - x = -2x, which is positive and non-zero. Thus, the domain is all negative real numbers, (-∞, 0). Option B and C include values where the denominator becomes zero.