Solve the equation: $|x-| 3 x+1||=4$.
(1994, Tianjin City Junior High School Mathematics Competition)
Analysis: This is an equation involving absolute values. Absolute values are defined by classification, so we should discuss three cases: $3 x+1=0$, $3 x+1>0$, and $3 x+1<0$.
(1) When $x=-\frac{1}{3}$, the left side of the equation $=\frac{1}{3}$, the right side $=4$. Contradiction, no solution.
(2) When $x-\frac{1}{3}$, the original equation becomes
$$
|x-(3 x+1)|=4 \text{, }
$$

i.e. $\left\{\begin{array}{l}-2 x-1=4, \\ -2 x-1=-4 .\end{array}\right.$
Solving, we get $x=-\frac{5}{2}$ (discard), $x=\frac{3}{2}$.