Let's first simplify the left side of the equation:

4/9 × (1 12x - 3/8)

Convert the mixed number 1 12x to an improper fraction:

1 12x = 9/9 + 12x = 9/9 + 96x/9 = 105x/9

Now we have:

4/9 × (105x/9 - 3/8)
= 4/9 × (105x/9 - 27/72)
= 4/9 × (105x/9 - 27/72)
= 4/9 × (105x/9 - 27/72)
= 4/9 × (105x - 27)/9
= (4 × (105x - 27))/9
= (420x - 108)/9
= 420x/9 - 108/9
= 46.67x - 12

Now let's simplify the right side of the equation:

1 5/6 - 1 1/3x
= 11/6 - 4/3x
= (11 - 8x)/6

Therefore, the equation becomes:

46.67x - 12 = (11 - 8x)/6

Now we can solve for x:

46.67x - 12 = (11 - 8x)/6
Multiply both sides by 6 to get rid of the denominator:
280x - 72 = 11 - 8x
Add 8x and 72 to both sides:
288x = 83
Divide by 288:
x = 83/288 ≈ 0.2889

So the solution to the equation is x ≈ 0.2889.