Simplify the function

\[f(x) = \dfrac{(x-1)-(x-2)}{(x-1)(x-2)} + \dfrac{(x-2)-(x-3)}{(x-2)(x-3)} + \dfrac{(x-3)-(x-4)}{(x-3)(x-4)} + \dfrac{(x-4)-(x-5)}{(x-4)(x-5)}.\]

Think about the following questions. Some may be more helpful than others.

  • Can a common denominator be found?

  • What happens if we simplify the numerator? What might the next step be?

  • Look at the individual fractions, e.g. \(\dfrac{(x-1)-(x-2)}{(x-1)(x-2)}\). Can it be split into simpler fractions?

Is it possible to split each of the following fractions up into the sum of two fractions?

  • \(\dfrac{1}{(x-2)(x-3)}\)

  • \(\dfrac{7}{(x-1)(x-8)}\)

  • \(\dfrac{1}{(x-2)(x-8)}\)

How does the fraction \(\dfrac{(x-2)-(x-3)}{(x-2)(x-3)}\) help?

Do you think all these fractions can be written in the form \(\dfrac{A}{(x-p)} + \dfrac{B}{(x-q)}\)?