$2 \cdot 130$ The solution to the equation $\left|\sqrt{(x-2)^{2}}-1\right|=x$ is
(A) $\frac{3}{2}$.
(B) $\frac{1}{2}$.
(C) $\frac{3}{2}$ or $\frac{1}{2}$.
(D) No solution.
(China Sichuan Province Junior High School Mathematics Competition, 1992)
［Solution］Discuss the following cases:
(1) When $2 \leqslant x \leqslant 3$, the original equation can be simplified to $|x-2-1|=x$, then $3-x=x$, solving gives $x=\frac{3}{2}$, which contradicts the initial assumption, so it is discarded.
(2) When $x>3$, the original equation can be simplified to $|x-2-1|=x$, then $x-3=x$, which is not valid.
(3) When $1 \leqslant x<2$, the original equation can be simplified to $|2-x-1|=x$, then $x-1=x$, which is not valid.
(4) When $x<1$, the original equation becomes $|2-x-1|=x$, then $1-x=x$, solving gives $x=\frac{1}{2}$, which satisfies the original equation.
In summary, the original equation has only one solution $x=\frac{1}{2}$. Therefore, the answer is $(B)$.