Example 9. Solve the inequality

$$
\frac{\lg ^{2} x-3 \lg x+3}{\lg x-1}<1
$$
Solution. Let $y=\lg x$, then the given inequality takes the form

$$
\frac{y^{2}-3 y+3}{y-1}<1
$$

This inequality is equivalent to the inequality

$$
\frac{y^{2}-3 y+3}{y-1}-1<0 \Leftrightarrow \frac{y^{2}-4 y+4}{y-1}<0
$$

i.e., the inequality

$$
\frac{(y-2)^{2}}{y-1}<0
$$

The solution to the last inequality is the interval $-\infty<$ $<y<1$.

From this, we conclude that the given inequality is equivalent to the inequality

$$
\lg x<1
$$

the solution of which is the interval $0<x<10$.