To start, let's simplify the left side of the equation:

5/6 * (1/2x - 2/3) = 3x - 2

First, distribute the 5/6:

5/6 1/2x - 5/6 2/3 = 3x - 2

Multiply the fractions:

5/12x - 10/18 = 3x - 2

Now, simplify the fractions:

5/12x - 5/9 = 3x - 2

Find a common denominator for the fractions:

To find a common denominator for 12 and 9, we can use 36.

Multiply the numerator and denominator of 5/12x by 3 and multiply the numerator and denominator of 5/9 by 4/4:

15/36x - 20/36 = 3x - 2

Now we have:

15x/36 - 20/36 = 3x - 2

Combine like terms:

(15x - 20)/36 = 3x - 2

Now, set the equation equal to 1/4:

(15x - 20)/36 = 3x - 2/4

Simplify the right side:

(15x - 20)/36 = 3x - 1/2

Multiply both sides by 36 to get rid of the denominator:

15x - 20 = 36(3x - 1/2)

15x - 20 = 108x - 18

Bring like terms to one side:

-20 + 18 = 108x - 15x

-2 = 93x

Divide by 93:

x = -2/93

Therefore, the solution to the equation 5/6(1/2x - 2/3) = 3x - 2, 1/4 is x = -2/93.