6.016. $\frac{x^{2}+1}{x}+\frac{x}{x^{2}+1}=2.9$.
## Solution.

Domain of definition: $x \neq 0$.

$$
\begin{aligned}
& \text { Let } \frac{x^{2}+1}{x}=y, \text { then } y+\frac{1}{y}-2.9=0 \Leftrightarrow \frac{y^{2}-2.9 y+1}{y}=0 \Leftrightarrow \\
& \Leftrightarrow\left\{\begin{array}{l}
y^{2}-2.9 y+1=0, \\
y \neq 0
\end{array} \Rightarrow y_{1}=\frac{5}{2}, y_{2}=\frac{2}{5} .\right.
\end{aligned}
$$

From this, $\frac{x^{2}+1}{x}=\frac{5}{2}$ or $\frac{x^{2}+1}{x}=\frac{2}{5}$.

The first equation has roots $x_{1}=2, x_{2}=\frac{1}{2}$, while the second equation has no solutions $(D<0)$.

Answer: $x_{1}=2, x_{2}=\frac{1}{2}$.