This is an expression, not an equation. However, the expression can be expressed as a partial fraction decomposition.
The denominator factors as ( 2x -1) (x + 2) (x -2). {Factoring by groups works here}
This means that the expression can be written in the form
A/(2 x -1) + B/(x + 2) + C/(x-2) for a set of numbers A, B, C. The remaining task is to find the values of A B and C.
This is done by adding fractions. The common denominator is (2x -1) (x +2) ( x-2) = 2 x3 - x2 -8 x +4
After working out the numerator resulting from the addition of fractions and grouping like powers of x,
the numerator becomes
x2 (2A + 2 B + C) + x( 3A - 5 B) (- 2 A + 2 B - 4C)
To get the original expression, it is required that
2A + 2 B + C = 9
3A -5B = -9 and
-2A + 2B - 4 C = 6
This set of three equations and three unknowns can be solved by standard techniques. The easiest is to use a calculator (such as the TI-84) which implements the reduced row echelon form ( rref ) function. The result is
A = 2, B = 3, C = -1
Thus the final answer is 2/( 2x + 1) + 3/(x +2) - /(x -2)
