We want to solve for x in:

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

To do so, we first multiply everything by the least common denominator: In this case that would be `3x (x+6)` . The denominator 3 is already factored in here since `3x` is a common denomintor of the first fraction and the last term.

Multiplying this out:

`2(x+6) = 8(3x) - 2(x)(x+6)`

Then, we have to expand and multiply out everything to get:

`2x + 12 = 24x - 2x^2 - 12x`

`2x^2 -10x + 12 = 0`

Dividing both sides by 2:

`x^2 - 5x + 6 = 0`

This is a quadratic equation that is easily factorable to:

`(x - 2)(x-3) = 0`

Using the zero product property:

`x = 2` or `x = 3` .

Hence, the answers are x = 2 or x = 3.

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TO check, we simply substitute:

If x = 2:

`2/(3*2) = 8/(2+6) - 2/3`

`2/6 = 8/8 - 2/3`

`1/3 = 1 - 2/3`

`1/3 = 1/3` which is true.

If x = 3:

`2/(3*3) = 8/(3+6) - 2/3`

`2/9 = 8/9 - 2/3`

`2/9 = 8/9 - 6/9`

`2/9 = 2/9` which is true!