1. The problem is to solve the equation $9.0 = 9.6 + \log \frac{x}{0.1 - x}$ for $x$. 2. Start by isolating the logarithmic term: $$9.0 - 9.6 = \log \frac{x}{0.1 - x}$$ $$-0.6 = \log \frac{x}{0.1 - x}$$ 3. Recall that $\log a = b$ means $a = 10^b$. So, $$\frac{x}{0.1 - x} = 10^{-0.6}$$ 4. Calculate $10^{-0.6}$: $$10^{-0.6} = \frac{1}{10^{0.6}} \approx 0.2512$$ 5. Substitute back: $$\frac{x}{0.1 - x} = 0.2512$$ 6. Cross-multiply: $$x = 0.2512 (0.1 - x)$$ 7. Distribute: $$x = 0.02512 - 0.2512x$$ 8. Add $0.2512x$ to both sides: $$x + 0.2512x = 0.02512$$ $$1.2512x = 0.02512$$ 9. Divide both sides by $1.2512$: $$x = \frac{0.02512}{1.2512}$$ 10. Simplify the fraction: $$x = \frac{\cancel{0.02512}}{\cancel{1.2512}} = 0.02008$$ 11. Final answer: $$\boxed{x \approx 0.0201}$$