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 you isolate the absolute value and a negative number is left on the other side, stop. The equation has no solution, and splitting it would give answers that do not check.

For example, in 2|x| + 3 = 1, subtract 3 and divide by 2 to get |x| = -1. There is no solution.

The equation |x - 4| = 0 has exactly one solution. The only number zero steps from 4 is 4 itself. Splitting gives x - 4 = 0 twice, because 0 and -0 are the same number, so the only solution is x = 4. The number of solutions depends on the sign of the number the absolute value equals: positive gives two, zero gives one, and negative gives none.