Example 2 Solve the equation $3 x^{3}-[x]=3$

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Example 2 Solve the equation $3 x^{3}-[x]=3$
Solve: First, estimate $x$: $|x|$ cannot be too large. If $|x| \geqslant 2$, then by property 2, we have
$$
\begin{aligned}
\left|3 x^{3}-[x]\right| & \geqslant 3|x|^{3}-|x|-1=|x|\left(3|x|^{2}-1\right)-1 \\
& \geqslant 2\left(3|x|^{2}-1\right)-1 \geqslant 2\left(3 \times 2^{2}-1\right)-1>3
\end{aligned}
$$

Therefore, $x \in(-2,2)$.
When $x \in(-2,-1)$, $[x]=-2$, the original equation becomes $3 x^{3}+2=3, 3 x^{3}=1$, which is discarded; when $x \in[-1,0)$, $[x]=-1$, the original equation becomes $3 x^{3}+1=3, 3 x^{3}=2$, which is discarded; when $x \in[0,1)$, $[x]=0$, the original equation becomes $3 x^{3}=3$, which is discarded; when $x \in[1,2)$, the original equation becomes $3 x^{3}-1=3, x^{3}=\frac{4}{3}, x=\sqrt[3]{\frac{4}{3}}$, which is a solution to the original equation.