Example 3. Solve the inequality

$$
x \log _{1 / 10}\left(x^{2}+x+1\right)>0
$$
Solution. Since $\log _{1 / 10} A=-\lg A$, the given inequality is equivalent to the inequality

$$
x \lg \left(x^{2}+x+1\right) 0 , } \\
{ \operatorname { l g } ( x ^ { 2 } + x + 1 ) 0
\end{array}\right.\right.
$$

Let's solve the first system. Since $x^{2}+x+1>1$ for $x>0$, then $\lg \left(x^{2}+x+1\right)>0$ for $x>0$; therefore, the first system of (1) has no solutions.

The second system of (1) is equivalent to the system

$$
\left\{\begin{array} { l } 
{ x  1 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
x0
\end{array} \Leftrightarrow x<-1\right.\right.
$$

Therefore, the interval $-\infty<x<-1$ is the set of all solutions to the original inequality.