Finding all real functions satisfying $f(x) + f \left( \frac{x - 1}{x} \right) = \frac{5x^2 - x - 5}{x}$ 
Find all functions $f:\mathbb R \to \mathbb R$ that satisfy
  $$f(x) + f \left( \frac{x - 1}{x} \right) = \frac{5x^2 - x - 5}{x}$$
  for all nonzero $x$.


We set up a system of $3$ functional equations and one of the variables will be $f(x)$ and we will solve for it. First replace $x$ by $\frac{1}{1-x}$ and we get: $f(\frac{1}{1-x}) + f(x) = P(x)$, with $P(x)$ is the right hand side evaluated at $\frac{1}{1-x}$. Next replace $x$ by $\dfrac{x-1}{x}\implies f(\frac{x-1}{x})+f(\frac{1}{1-x}) = Q(x)$, with $Q(x)$ is the right hand side evaluated at $\frac{x-1}{x}$. Combine with the original equation: $f(x) + f(\frac{x-1}{x}) = \dfrac{5x^2-x-5}{x}$, you do have a system of "linear" equations in $3$ variables each of which is a function and your $f(x)$ is one of them, and you can solve for $f(x)$ by elimination method in a standard college algebra course.
