First, we need to find the LCD (Least Common Denominator) of the fractions on both sides of the equation. The denominators of the fractions are x^2 - 7x + 12, x - 3, and x - 4.
The factorization of the denominator of the first fraction is:
x^2 - 7x + 12 = (x - 4)(x - 3)
So, the LCD is (x - 4)(x - 3).
We can now rewrite the equation with the LCD:
5/(x - 4)(x - 3) - 2/(x - 3) = 5/(x - 4)
Multiplying both sides by the LCD, we get:
5 - 2(x - 4) = 5(x - 3)
Simplifying:
5 - 2x + 8 = 5x - 15
Collecting like terms:
7x = 28
x = 4
We have found that x = 4 is the solution to the equation.
To check our solution, we need to verify that it does not produce any denominators equal to zero.
The original equation with x = 4 is:
5/4^2 - 7(4) + 12 - 2/4 - 3 = 5/4 - 4
Simplifying:
5/4 - 4 = 5/4 - 4
Therefore, the solution x = 4 is valid.