The problem is to solve the equation $\sqrt{x+8} + \sqrt{x+1} = 7$.

1. Problem Description

The problem is to solve the equation $\sqrt{x+8} + \sqrt{x+1} = 7$.

2. Solution Steps

To solve the equation $\sqrt{x+8} + \sqrt{x+1} = 7$, we first isolate one of the square roots.

$\sqrt{x+8} = 7 - \sqrt{x+1}$

Now, square both sides of the equation:

$(\sqrt{x+8})^2 = (7 - \sqrt{x+1})^2$

$x+8 = 49 - 14\sqrt{x+1} + (x+1)$

$x+8 = 49 - 14\sqrt{x+1} + x + 1$

$x+8 = x + 50 - 14\sqrt{x+1}$

Subtract $x$ from both sides:

$8 = 50 - 14\sqrt{x+1}$

Subtract 50 from both sides:

$-42 = -14\sqrt{x+1}$

Divide both sides by -14:

$3 = \sqrt{x+1}$

Now, square both sides again:

$3^2 = (\sqrt{x+1})^2$

$9 = x+1$

Subtract 1 from both sides:

$x = 8$

Now, we verify the solution by substituting $x=8$ into the original equation:

$\sqrt{8+8} + \sqrt{8+1} = \sqrt{16} + \sqrt{9} = 4 + 3 = 7$.

Since the equation holds true, the solution is valid.

3. Final Answer

The solution is $x = 8$.