No Solution, One Solution

An absolute value is never negative, so $\left|x\right| = -2$ 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 $2\left|x\right|+3 = 1$, subtract 3 and divide by 2 to get $\left|x\right| = -1$. There is no solution.

The equation $\left|x-4\right| = 0$ 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.