To solve this equation, we first need to simplify the left side by combining like terms.

2/3x - 1/9 - (4 + 1/6x)

First we distribute the negative sign inside the parentheses:

2/3x - 1/9 - 4 - 1/6x

Now we combine like terms:

2/3x - 1/6x - 1/9 - 4

To combine the x terms, we need a common denominator. The least common multiple of 3 and 6 is 6, so we multiply the first term by 2/2:

4/6x - 1/6x - 1/9 - 4

Now we can combine the x terms:

(4/6 - 1/6)x - 1/9 - 4

This simplifies to:

(3/6)x - 1/9 - 4
(1/2)x - 1/9 - 4
(1/2)x - 37/9

Now we can rewrite the equation with the simplified left side:

(1/2)x - 37/9 = 2/9 - 5/6x

Next, we need to isolate the x terms on one side of the equation. Let's add 5/6x to both sides:

(1/2)x + 5/6x - 37/9 = 2/9

To combine the x terms, we need a common denominator. The least common multiple of 2 and 6 is 6, so we multiply the first term by 3/3:

(3/6)x + 5/6x - 37/9 = 2/9

Now we can combine the x terms:

(3/6 + 5/6)x - 37/9 = 2/9
(8/6)x - 37/9 = 2/9
(4/3)x - 37/9 = 2/9

Next, we can isolate the x term by moving the constant terms to the other side:

(4/3)x = 2/9 + 37/9
(4/3)x = 39/9
(4/3)x = 13/3

Finally, we can solve for x by multiplying both sides by 3/4:

x = (13/3) * (3/4)
x = 13/4
x = 3.25

Therefore, the solution to the equation is x = 3.25.