G4.2 Suppose there are $b$ distinct solutions of the equation $|x-| 2 x+1||=3$, find the value of $b$.
$$
\begin{array}{l}
|x-| 2 x+1||=3 \\
x-|2 x+1|=3 \text { or } x-|2 x+1|=-3 \\
x-3=|2 x+1| \text { or } x+3=|2 x+1| \\
x-3=2 x+1 \text { or } 3-x=2 x+1 \text { or } x+3=2 x+1 \text { or } 2 x+1=-x-3 \\
x=-4 \text { or } \frac{2}{3} \text { or }-2 \text { or }-\frac{4}{3}
\end{array}
$$

Check: when $x=-4$ or $\frac{2}{3}, x-3=|2 x+1| \geq 0$, no solution
When $x=-2$ or $-\frac{4}{3}, x+3=|2 x+1| \geq 0$, accepted
There are 2 real solutions.