To solve this equation, we first need to combine like terms and simplify the expression:

4/9x^2 + x/3x - 1 = 4/3x + 1

To combine the x terms in the numerator, we need to have a common denominator. So, the second term becomes:

(x/3x) = (x^2)/3x^2

Now, our equation becomes:

4/9x^2 + (x^2)/3x - 1 = 4/3x + 1

Next, we need to get rid of the fractions by multiplying the entire equation by the least common multiple of the denominators (9x^2):

9x^2(4/9x^2) + 9x^2(x^2/3x) - 9x^2(1) = 9x^2(4/3x) + 9x^2(1)

This simplifies to:

4 + 3x - 9x^2 = 12 + 9x^2

Rearranging the terms, we get:

18x^2 + 3x - 8 = 0

Now, we have a quadratic equation. We can solve for x by factoring or using the quadratic formula:

x = [-b ± sqrt(b^2 - 4ac)] / 2a

Plugging in the values of a = 18, b = 3, and c = -8 into the formula, we get:

x = [-3 ± sqrt(3^2 - 4 18 -8)] / 2 * 18
x = [-3 ± sqrt(9 + 576)] / 36
x = [-3 ± sqrt(585)] / 36
x = [-3 ± sqrt(585)] / 36

Therefore, the solution to the equation is x ≈ -0.188 or x ≈ 0.421.