To solve this equation, we will first isolate the square root term on one side of the equation.

(sqrt x) - 1 = sqrt( x - (sqrt x + 7) - 1)

Square both sides to eliminate the square root:

((sqrt x) - 1)^2 = (sqrt( x - (sqrt x + 7) - 1))^2

x - 2(sqrt x) + 1 = x - (sqrt x + 7) - 1

Expand both sides:

x - 2(sqrt x) + 1 = x - sqrt x - 7 - 1

Combine like terms:

-2(sqrt x) + 1 = -sqrt x - 8

Add sqrt x to both sides:

-sqrt x - 2(sqrt x) + 1 = - 8

-3(sqrt x) + 1 = -8

Subtract 1 from both sides:

-3(sqrt x) = -9

Divide by -3:

sqrt x = 3

Square both sides to get x:

x = 9

Therefore, the solution to the given equation is x = 9.