Example 3 Solve the inequality $(\lg x+3)^{7}+\lg ^{7} x+\lg x^{2}+3 \geqslant 0$.

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Solve $(\lg x+3)^{7}+(\lg x+3) \geqslant-\lg ^{7} x-\lg x$.

The function $f(t)=t^{7}+t$ is an odd function and is monotonically increasing on $(-\infty,+\infty)$, therefore
$$
\lg x+3 \geqslant-\lg x, \lg x \geqslant-\frac{3}{2}, x \geqslant 10^{-\frac{3}{2}} .
$$

Thus, the solution set of the inequality is $\left[10^{-\frac{3}{2}},+\infty\right)$.