$1 \cdot 2$ If $x<-2$, then $1-|1+x|$ equals
(A) $2+x$.
(B) $-2-x$.
(C) $x$.
(D) $-x$.
(E) -2.
(25th American High School Mathematics Examination, 1974)
[Solution] By definition $|a|=\left\{\begin{aligned} a, & a \geqslant 0, \\ -a, & a<0 .\end{aligned}\right.$ If $x<-2$, then $1+x<0$, so
$$
\begin{array}{c}
|1+x|=-(1+x), \\
\therefore \quad|1-| 1+x||=|1+1+x|=|2+x| .
\end{array}
$$

Also, if $x<-2$, then $2+x<0$, so $|2+x|=-2-x$. Therefore, the answer is $(B)$.