Let's consider part (b) first. Since we have a form for the inverse of adding fractions, we apply the direct process (addition) to the form, and try to match the answer to the expression with which we started. If we write the sum `A/x+B/(c-x) ` with a common denominator, we obtain
In order for this to equal
the numerators must match. Thus we must have
`Atext[(]c-x text[)]+Bx=1` for all values of `x`.
To find algebraic equations that will pin down `A` and `B`, we may substitute any values we want for `x`. Particularly nice values to substitute are `x=0` and `x=c`. For `x=0`, we obtain `Ac=1`, or `A=1//c`. And for `x=c`, we have `Bc=1`, so `B` is also `1//c.` In particular, in part (a) you should have found that both `A` and `B` are `1//2`.