Question

The rational function (x-4)/x(2x-1) can be expressed as the sum of two partial fractions: A/x and B/(2x-1). Find the value of A – B.
a) -11
b) 11
c) -3
d) 3

Answer 1

a/x + b/(2x-1) = (x-4) / [ x(2x-1) ] I assume you mean
multiply both sides by [x(2x-1)]

a(2x-1) + b x = x-4
2ax - a + bx = 1x - 4
(2a+b)x = 1x
-a = -4
so a = 4
(8+b ) = 1
b = -7

Answer 2

4 - (-7) = 11

Answer 3

Let (x-4)/(x(2x-1)) = A/x + B/(2x-1) = [ A(2x-1) + Bx]/(x(2x-1))
(how nice of them to factor the denominator for us)

x - 4 = A(2x-1) + Bx
This becomes an identity, thus true for all values of x
let x = 0 -----> -4 = -A or A = 4
let x = 1/2 ----> -7/2 = B/2 or B = -7

check: 4/x - 7/(2x-1) = (4(2x-1) - 7x)/(x(2x-1)) = (x - 4)/(x(2x-1))

so they want B-A = ....