2.81 The number of real solutions to the equation $|x-2|+|x-3|=1$ is
(A) 0.
(B) 1.
(C) 2.
(D) 3.
(E) More than 3.
(41st American High School Mathematics Examination, 1990)
[Solution] Consider three segments: $x<2$, $2 \leqslant x \leqslant 3$, and $x \geqslant 3$.
When $2 \leqslant x \leqslant 3$, the equation simplifies to $(x-2)+(3-x)=1$, which is $1=1$, an identity. Therefore, the equation has infinitely many solutions in this interval.
Hence, the answer is $(E)$.