1. Definición de variables:
Sea $A = \arcsin \frac{2}{3\sqrt{x}}$ y $B = \arcsin \sqrt{1-x}$. La ecuación es $A - B = \arcsin(1/3)$.
Aplicamos la función seno a ambos lados:
$$ \sin(A - B) = \sin(\arcsin(1/3)) = 1/3 $$
$$ \sin A \cos B - \cos A \sin B = 1/3 $$

2. Desarrollo paso a paso:
$\sin A = \frac{2}{3\sqrt{x}}$, $\cos A = \sqrt{1 - \frac{4}{9x}}$.
$\sin B = \sqrt{1-x}$, $\cos B = \sqrt{1 - (1-x)} = \sqrt{x}$.
Sustituyendo:
$$ \begin{aligned} \frac{2}{3\sqrt{x}} (\sqrt{x}) - \left(\sqrt{\frac{9x-4}{9x}}\right)(\sqrt{1-x}) &= \frac{1}{3} \\ \frac{2}{3} - \frac{\sqrt{9x-4}\sqrt{1-x}}{3\sqrt{x}} &= \frac{1}{3} \\ \frac{1}{3} &= \frac{\sqrt{(9x-4)(1-x)}}{3\sqrt{x}} \\ \sqrt{x} &= \sqrt{-9x^2 + 13x - 4} \end{aligned} $$
Elevando al cuadrado:
$$ x = -9x^2 + 13x - 4 \implies 9x^2 - 12x + 4 = 0 $$
Factorizando:
$$ (3x - 2)^2 = 0 \implies 3x = 2 $$
$$ \boxed{x = \frac{2}{3}} $$