How do I solve the rational equation (2/3x)+(2/3)=(8/(x+6)) ?

[2/(3x)]+(2/3)] = [8/(x+6)]
[2/(3x)]+(2/3)] * 3x * (x + 6) * (1/2) = [8/(x+6)] * 3x * (x + 6) * (1/2)
(x + 6) + x(x + 6) = 4 * 3x
x + 6 + x² + 6x = 12x
x² - 5x + 6 = 0
(x - 2)(x - 3) = 0
x = 2 or x = 3

This may be another order-of-operations issue. We see a lot of that here. I will give you the benefit of the doubt, and assume that you mean just what you wrote. This is a literal interpretation.

(2/3x)+(2/3)=(8/(x+6))
2/3x + 2/3 = 8/(x + 6) ... removing superfluous brackets

Multiply by 3(x + 6). That gets rid of the fractions, at least for now.

2x(x + 6) + 2(x + 6) = 24
2x² + 12x + 2x + 12 = 24
2x² + 14x - 12 = 0
x² + 7x - 6 = 0
x = [-7 ± √(73)]/2

Multiply all terms and both sides of equal by (x+6)

(x+6)(2/3x) +(x+6)(2/3) = (8/(x+6)) (x+6)

Now distribute, combine like items and solve:

2/3 x² + 4x + 2/3x + 4 = 8
multiply by 3
2 x² + 12x + 2x + 12 = 24

combine
2x² + 14x -12 = 0

divide by 2

x² + 7x -6 = 0

When you have the equation in y = ax² + bx + c =0 form, then:
Quadratic Formula is x= -b/2a ±√(b² -4ac)/2a

I prefer this in this form since it's written as the 'vertex' ± intercepts

x = - 7/2 ± ½ sqrt(73)

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By the way, you might get 2 totally different answers to this question. Why?
The first term (2/3x) can be interpreted as (⅔ x) or 2x/3, or as 2/(3x) as used by 不用客氣 below.
The value I used was the accepted method per PEMDAS, and verified by WolframAlpha (see image):

To prevent misunderstandings, you must insert critical parens to ensure we have the right problem.

(2/3 x) + (2/3) = (8 / (x + 6))
(2 x) + (2) = (24 / (x + 6))
(2 x)(x + 6) + (2)(x + 6) = 24
2x^2 + 14x -12 = 0
2(x^2 + 7x - 6) = 0
Solutions:
x = -7/2 - sqrt(73)/2
x = sqrt(73)/2 - 7/2

(2/3x)+ (2/3) =(8/(x+6))--->
2/3 ( (1/x) +1) = 8(1/(x+6)--->
(2/3)/8 = (1/(x+6))/ ((1/x)+1)--->
(2/3)(1/8)= 1/(x+6) * 1/((1/x) +1)--->
2/24 = 1/(x+6) * 1/((1/x) +1) --->
1/12= 1/(x+6) * 1/((1/x)+1)--->
12 = (x+6) * (1/x+1)--->
12= x +6/x +7--->
5= x+6/X--->
x= 2

Answer: x=2

(2/3x)+(2/3)=(8/(x+6)) <--->(1/3x)+(1/3)=(4/(x+6))
LCD=3x(x+6)
(x+6)+x(x+6)=12x
solve for x