$4 \cdot 33$ Find all real numbers that satisfy the equation $|x+3|-|x-1|=x+1$.

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［Solution］When $x \leqslant-3$, the equation becomes
$$
-(x+3)-(1-x)=x+1 \text {, }
$$

which yields $x=-5$.
When $-3<x \leqslant 1$, the equation becomes
$$
(x+3)-(1-x)=x+1,
$$

which simplifies to
$$
x=-1 \text {. }
$$
When $x>1$, the equation becomes
$$
(x+3)-(x-1)=x+1,
$$

which yields
$$
x=3 \text {. }
$$

Therefore, the solutions are $-5,-1,3$.