1. **State the problem:** Simplify and analyze the expression $$x + \frac{1}{3x - 5} - \frac{8}{3} x \sqrt{8 + \frac{9x - 5}{24}}.$$\n\n2. **Rewrite the expression:**\n$$x + \frac{1}{3x - 5} - \frac{8}{3} x \sqrt{8 + \frac{9x - 5}{24}}.$$\n\n3. **Simplify inside the square root:**\nCalculate the term inside the square root:\n$$8 + \frac{9x - 5}{24} = \frac{8 \times 24}{24} + \frac{9x - 5}{24} = \frac{192 + 9x - 5}{24} = \frac{187 + 9x}{24}.$$\n\n4. **Rewrite the expression with simplified root:**\n$$x + \frac{1}{3x - 5} - \frac{8}{3} x \sqrt{\frac{187 + 9x}{24}}.$$\n\n5. **Simplify the square root:**\n$$\sqrt{\frac{187 + 9x}{24}} = \frac{\sqrt{187 + 9x}}{\sqrt{24}} = \frac{\sqrt{187 + 9x}}{2\sqrt{6}}.$$\n\n6. **Substitute back:**\n$$x + \frac{1}{3x - 5} - \frac{8}{3} x \times \frac{\sqrt{187 + 9x}}{2\sqrt{6}} = x + \frac{1}{3x - 5} - \frac{8x \sqrt{187 + 9x}}{3 \times 2 \sqrt{6}} = x + \frac{1}{3x - 5} - \frac{4x \sqrt{187 + 9x}}{3 \sqrt{6}}.$$\n\n7. **Final simplified form:**\n$$\boxed{x + \frac{1}{3x - 5} - \frac{4x \sqrt{187 + 9x}}{3 \sqrt{6}}}.$$\n\nThis is the simplified expression. Further solving depends on the context (e.g., solving for $x$ or graphing).