1. **State the problem:** Solve the equation $$\sqrt{x + 1} = \sqrt{x + 8}$$ for all values of $x$. 2. **Recall the property of square roots:** If $$\sqrt{A} = \sqrt{B}$$, then $$A = B$$ provided both sides are defined (i.e., $A \geq 0$ and $B \geq 0$). 3. **Apply the property:** Set the expressions inside the square roots equal: $$x + 1 = x + 8$$ 4. **Simplify the equation:** $$x + 1 = x + 8$$ Subtract $x$ from both sides: $$\cancel{x} + 1 = \cancel{x} + 8$$ $$1 = 8$$ 5. **Analyze the result:** The statement $1 = 8$ is false, which means there is no solution to the equation. 6. **Check the domain:** Both $\sqrt{x + 1}$ and $\sqrt{x + 8}$ require $x + 1 \geq 0$ and $x + 8 \geq 0$, so $x \geq -1$. 7. **Conclusion:** Since the equation leads to a contradiction, there are no values of $x$ that satisfy the equation. **Final answer:** No solution.