Hi there!
[tex]\large\boxed{x = -1}[/tex]
We can solve by multiplying each term to get a common denominator:
[tex]\frac{1}{x - 4} + \frac{x}{x - 2} = \frac{2}{x^2-6x+8}[/tex]
[tex]\frac{x -2}{(x - 4)(x - 2)} + \frac{x(x - 4)}{(x - 2)(x - 4)} = \frac{2}{x^2-6x+8}[/tex]
Since the denominators are all the same (x² - 6x + 8 factors to (x - 2)(x - 4)), we can ignore them temporarily and solve the numerator:
x - 2 + x(x - 4) = 2
Simplify:
x - 2 + x² - 4x = 2
x² - 3x - 4 = 0
Factor:
(x - 4)(x + 1) = 0
x = -1, 4
For the values of x to be a solution, they cannot cause the denominator to be equal to 0, so:
(x - 2)(x - 4) = 0
x = 2, 4. These values result in a denominator of 0, which is undefined.
Thus, the only solution is x = -1.
