1. **State the problem:** Solve for all values of $x$ in the equation $$\sqrt{x + 1} = \sqrt{x + 8}.$$\n\n2. **Understand the equation:** Both sides are square roots. For the equation to hold, the expressions inside the roots must be equal or the roots must be equal in value.\n\n3. **Square both sides to eliminate the square roots:**\n$$\left(\sqrt{x + 1}\right)^2 = \left(\sqrt{x + 8}\right)^2$$\nwhich simplifies to\n$$x + 1 = x + 8.$$\n\n4. **Simplify the equation:**\n$$x + 1 = x + 8$$\nSubtract $x$ from both sides:\n$$\cancel{x} + 1 = \cancel{x} + 8$$\nwhich gives\n$$1 = 8,$$\nwhich is false.\n\n5. **Interpretation:** Since the simplified equation is false, there is no value of $x$ that satisfies the original equation.\n\n6. **Check domain:** The expressions under the square roots must be non-negative:\n$$x + 1 \geq 0 \Rightarrow x \geq -1,$$\n$$x + 8 \geq 0 \Rightarrow x \geq -8.$$\nThe domain is $x \geq -1$.\n\n7. **Conclusion:** No $x$ in the domain satisfies the equation, so there is no solution.\n\n**Final answer:** No solution.