4.33 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
$$-(x+3)-(1-x)=x+1$$

which yields $\square$
$$x=-5 .$$

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

which yields $\square$
$$x=-1 .$$

When $x > 1$, the equation becomes
$$(x+3)-(x-1)=x+1$$

which yields $\square$
$$x=3 .$$

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