1. **State the problem:** Solve the equation $$5 \left(\frac{x}{3}\right)^2 - 3 \left(\frac{1}{9}\right) = - \frac{8}{9}$$ for $x$. 2. **Recall the formula and rules:** We will simplify and isolate $x^2$ step by step. Remember that squaring a fraction means squaring numerator and denominator separately: $$\left(\frac{a}{b}\right)^2 = \frac{a^2}{b^2}$$. 3. **Simplify the terms:** $$5 \left(\frac{x}{3}\right)^2 = 5 \cdot \frac{x^2}{9} = \frac{5x^2}{9}$$ $$3 \left(\frac{1}{9}\right) = \frac{3}{9} = \frac{1}{3}$$ 4. **Rewrite the equation:** $$\frac{5x^2}{9} - \frac{1}{3} = - \frac{8}{9}$$ 5. **Add $\frac{1}{3}$ to both sides to isolate the $x^2$ term:** $$\frac{5x^2}{9} = - \frac{8}{9} + \frac{1}{3}$$ 6. **Find a common denominator on the right side:** $$\frac{1}{3} = \frac{3}{9}$$ So, $$- \frac{8}{9} + \frac{3}{9} = - \frac{5}{9}$$ 7. **Now the equation is:** $$\frac{5x^2}{9} = - \frac{5}{9}$$ 8. **Multiply both sides by 9 to clear denominators:** $$\cancel{9} \cdot \frac{5x^2}{\cancel{9}} = -5$$ which simplifies to $$5x^2 = -5$$ 9. **Divide both sides by 5:** $$\frac{\cancel{5}x^2}{\cancel{5}} = \frac{-5}{5}$$ which simplifies to $$x^2 = -1$$ 10. **Interpret the result:** Since $x^2 = -1$ and the square of a real number cannot be negative, there is no real solution. **Final answer:** No real solution exists for $x$ because $x^2 = -1$ is impossible in real numbers.