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 \(\dfrac{A}{(x-p)} + \dfrac{B}{(x-q)}\)?
