1. **State the problem:** We need to find values of $x$ that cannot be solutions to the equation $$\frac{1}{x - 3} - \frac{1}{x + 8} = 10.$$ These values are those that make the equation undefined. 2. **Identify restrictions:** The equation involves denominators $x - 3$ and $x + 8$. Division by zero is undefined, so values that make any denominator zero cannot be solutions. 3. **Set denominators equal to zero:** $$x - 3 = 0 \implies x = 3$$ $$x + 8 = 0 \implies x = -8$$ 4. **Conclusion:** The values $x = 3$ and $x = -8$ make the denominators zero, so they cannot be solutions. **Final answer:** The values that cannot be solutions are $\boxed{-8, 3}$.