No Solution, One Solution

An absolute value is never negative, so |x| = -2 has no solution: no number is minus two steps from zero. When isolating the bars leaves a negative on the other side, stop there; the equation has no solution, and splitting it would invent answers that do not check.

2|x| + 3 = 1: subtract 3, divide by 2, and |x| = -1 appears. No solution.

And |x - 4| = 0 has exactly one: the only number zero steps from 4 is 4 itself. Splitting gives x - 4 = 0 twice (0 and -0 are the same number), so x = 4 once. Two solutions is the usual count; zero and one are the two special cases, decided by the sign of the number the bars equal.