The correct statements are:
There is one acceptable solution to the equation.
6 is an extraneous solution because it creates a 0 in the denominator.
x/(x - 2) = 4/(x² - 8x + 12) - 1/(x - 6)
x/(x - 2) = 4/(x - 2)(x - 6) - 1/(x - 6) {factorize the quadratic expression}
x/(x - 2) = [4 - (x - 2)]/(x - 2)(x - 6)
x/(x - 2) = (6 - x)/(x - 2)(x - 6)
x(x - 6) = 6 - x {cross multiplication}
x² - 6x + x - 6 = 0 {simplify and equate to 0}
(x + 1)(x - 6) = 0
x = -1 or x = 6.
Acceptable Solution: Upon solving the given equation, it's found that x = -1 is a valid solution. By substituting x = -1 into the equation, both sides equate, satisfying the equation.
Extraneous Solution: When x = 6, the equation leads to a division by zero, making the solution x = 6 extraneous. This happens because substituting x = 6 results in a denominator of zero in one of the terms, violating the fundamental rule of avoiding division by zero.
In summary, the equation has one valid solution, x = -1, while x = 6 is an extraneous solution due to division by zero. Therefore, the first statement is true, and statement fourth is also true.