Food for thought

Suggestion

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