1. **State the problem:** Solve the equation $$0.6 = \frac{(1 + \frac{x}{2})^2}{(1 - x)^2}$$ for $x$. 2. **Rewrite the equation:** We have $$0.6 = \frac{(1 + \frac{x}{2})^2}{(1 - x)^2}$$ which can be written as $$0.6 = \frac{\left(1 + \frac{x}{2}\right)^2}{(1 - x)^2}$$. 3. **Take the square root of both sides:** Since both numerator and denominator are squared, take the positive and negative square roots: $$\sqrt{0.6} = \pm \frac{1 + \frac{x}{2}}{1 - x}$$ 4. **Express the square root:** $$\sqrt{0.6} = \pm \frac{1 + \frac{x}{2}}{1 - x}$$ 5. **Set up two equations:** **Case 1:** $$\sqrt{0.6} = \frac{1 + \frac{x}{2}}{1 - x}$$ **Case 2:** $$-\sqrt{0.6} = \frac{1 + \frac{x}{2}}{1 - x}$$ 6. **Solve Case 1:** Multiply both sides by $(1 - x)$: $$\sqrt{0.6}(1 - x) = 1 + \frac{x}{2}$$ 7. **Distribute:** $$\sqrt{0.6} - \sqrt{0.6}x = 1 + \frac{x}{2}$$ 8. **Group $x$ terms on one side:** $$- \sqrt{0.6}x - \frac{x}{2} = 1 - \sqrt{0.6}$$ 9. **Factor out $x$:** $$x\left(- \sqrt{0.6} - \frac{1}{2}\right) = 1 - \sqrt{0.6}$$ 10. **Solve for $x$:** $$x = \frac{1 - \sqrt{0.6}}{- \sqrt{0.6} - \frac{1}{2}}$$ 11. **Simplify denominator:** $$- \sqrt{0.6} - \frac{1}{2} = -\left(\sqrt{0.6} + \frac{1}{2}\right)$$ 12. **Rewrite $x$:** $$x = \frac{1 - \sqrt{0.6}}{-\left(\sqrt{0.6} + \frac{1}{2}\right)} = - \frac{1 - \sqrt{0.6}}{\sqrt{0.6} + \frac{1}{2}}$$ 13. **Rationalize numerator and denominator if desired, or approximate:** $$\sqrt{0.6} \approx 0.7746$$ 14. **Calculate numerator:** $$1 - 0.7746 = 0.2254$$ 15. **Calculate denominator:** $$0.7746 + 0.5 = 1.2746$$ 16. **Calculate $x$:** $$x \approx - \frac{0.2254}{1.2746} = -0.1768$$ 17. **Solve Case 2:** $$-\sqrt{0.6} = \frac{1 + \frac{x}{2}}{1 - x}$$ Multiply both sides by $(1 - x)$: $$-\sqrt{0.6}(1 - x) = 1 + \frac{x}{2}$$ 18. **Distribute:** $$-\sqrt{0.6} + \sqrt{0.6}x = 1 + \frac{x}{2}$$ 19. **Group $x$ terms:** $$\sqrt{0.6}x - \frac{x}{2} = 1 + \sqrt{0.6}$$ 20. **Factor out $x$:** $$x\left(\sqrt{0.6} - \frac{1}{2}\right) = 1 + \sqrt{0.6}$$ 21. **Solve for $x$:** $$x = \frac{1 + \sqrt{0.6}}{\sqrt{0.6} - \frac{1}{2}}$$ 22. **Calculate denominator:** $$0.7746 - 0.5 = 0.2746$$ 23. **Calculate numerator:** $$1 + 0.7746 = 1.7746$$ 24. **Calculate $x$:** $$x \approx \frac{1.7746}{0.2746} = 6.46$$ **Final answers:** $$x \approx -0.177 \text{ or } x \approx 6.46$$