6.020. $\frac{x-a}{x-b}+\frac{x-b}{x-a}=2.5$.
## Solution.

Domain of definition: $x \neq b, x \neq a$.

$$
\text { Let } \frac{x-a}{x-b}=y: y+\frac{1}{y}-2.5=0 ; \frac{y^{2}-2.5 y+1}{y}=0
$$

The equation $y^{2}-2.5 y+1=0$ has roots $y_{1}=\frac{1}{2}, y_{2}=2$. We obtain

two equations in terms of $x: \frac{x-a}{x-b}=\frac{1}{2}$, from which $x_{1}=2 a-b$, or $\frac{x-a}{x-b}=2$, from which $x_{2}=2 b-a$.

Answer: if $a \neq b$, then $x_{1}=2 a-b, x_{2}=2 b-a$; if $a=b$, then there are no roots.