6.149. $\frac{x^{2}+1}{x}+\frac{x}{x^{2}+1}=-2.5$.
Solution.

Domain of definition: $x \neq 0$.

Let $\frac{x^{2}+1}{x}=y$. We get $y+\frac{1}{y}=-2.5 \Leftrightarrow y^{2}+2.5 y+1=0$, from which $y_{1}=-2, y_{2}=-\frac{1}{2}$. Then $\frac{x^{2}+1}{x}=-2$ or $\frac{x^{2}+1}{x}=-\frac{1}{2}$. From this, $x^{2}+2 x+1=0$ or $2 x^{2}+x+2=0$ for $x \neq 0$. Solving these equations, we get $x_{1,2}=-1$; the second equation has no roots, since $D<0$.

Answer: $x_{1,2}=-1$.