1. Stating the problem: Solve the equation $$\sqrt{x+8} + \sqrt{x+1} = 7$$. 2. Isolate one square root: Let’s isolate $$\sqrt{x+8}$$. $$\sqrt{x+8} = 7 - \sqrt{x+1}$$ 3. Square both sides to eliminate the square root on the left: $$\left(\sqrt{x+8}\right)^2 = \left(7 - \sqrt{x+1}\right)^2$$ $$x + 8 = 49 - 14\sqrt{x+1} + (x+1)$$ 4. Simplify the right side: $$x + 8 = 49 + x + 1 - 14\sqrt{x+1}$$ $$x + 8 = x + 50 - 14\sqrt{x+1}$$ 5. Subtract $$x$$ from both sides: $$8 = 50 - 14\sqrt{x+1}$$ 6. Subtract 50 from both sides: $$8 - 50 = -14\sqrt{x+1}$$ $$-42 = -14\sqrt{x+1}$$ 7. Divide both sides by $$-14$$: $$\frac{-42}{-14} = \sqrt{x+1}$$ $$3 = \sqrt{x+1}$$ 8. Square both sides to solve for $$x$$: $$3^2 = x + 1$$ $$9 = x + 1$$ 9. Subtract 1 from both sides: $$x = 8$$ 10. Verify the solution by plugging $$x = 8$$ back into the original equation: $$\sqrt{8+8} + \sqrt{8+1} = \sqrt{16} + \sqrt{9} = 4 + 3 = 7$$ The left side equals the right side, so $$x = 8$$ is a valid solution. Final answer: $$x = 8$$.