To solve for x, we need to first combine the fractions:

4/6 + 1/x = x-4

Next, find a common denominator for the fractions, which in this case would be 6x:

(4x + 6) / 6x + 1/x = x - 4

Now, multiply both sides of the equation by 6x to eliminate the fractions:

4x + 6 + 6 = 6x(x - 4)

Simplify:

4x + 12 = 6x^2 - 24x

Rearrange the equation into standard form:

6x^2 - 24x - 4x - 12 = 0

6x^2 - 28x - 12 = 0

Now, we can solve for x using the quadratic formula:

x = (-(-28) ± √((-28)^2 - 4(6)(-12))) / 2(6)

x = (28 ± √(784 + 288)) / 12

x = (28 ± √1072) / 12

x = (28 ± √(16*67)) / 12

x = (28 ± 4√67) / 12

x = (7 ± √67) / 3

So, the solutions for x are:

x = (7 + √67) / 3 or x = (7 - √67) / 3