.
I want to find the values of A and B such that this identity takes place for all x from the domain
=
+
. (1)
The right side is, after writing with the common denominator
. (2)
The denominator in (2) is the same as in the left side of (1).
Therefore, the necessary and sufficient condition for the identity to be valid is the equality
of the numerator of the left side of (1) and the numerator of (2).
So, this identuty must be held
Bx - 11 = Ax - 5A + 3x - 6.
It implies
B = A + 3, (3)
-11 = -5A - 6 (4)
From (4), -5A = -11 + 6 = -5; hence A = 1.
Then from (3), B = 1 + 3 = 4.
So, the answer is A + B = 1 + 4 = 5.
Solved.