1. **Stating the problem:** We are given the expression $\frac{\frac{2x}{9}}{1 - 2x} \div (1 - x - 3)$ and asked to solve it or simplify it. 2. **Understanding the expression:** The numerator is $\frac{2x}{9}$ divided by $1 - 2x$, which can be written as $\frac{\frac{2x}{9}}{1 - 2x} = \frac{2x}{9(1 - 2x)}$. The denominator is $1 - x - 3 = -x - 2$. So the entire expression is: $$\frac{\frac{2x}{9(1 - 2x)}}{-x - 2}$$ 3. **Simplifying the complex fraction:** Dividing by $-x - 2$ is the same as multiplying by its reciprocal: $$\frac{2x}{9(1 - 2x)} \times \frac{1}{-x - 2} = \frac{2x}{9(1 - 2x)(-x - 2)}$$ 4. **Final simplified form:** $$\boxed{\frac{2x}{9(1 - 2x)(-x - 2)}}$$ This is the simplified expression. 5. **Note on the integral symbol:** The integral symbol shown in the problem statement seems unrelated to the algebraic fraction simplification. Since no integral limits or integrand are provided, we focus on simplifying the given algebraic expression.