To solve the equation, we can start by isolating one of the square roots on one side of the equation. Let's isolate the square root of (x-1):

√(x-1) = 3 - √(3x-2)

Next, square both sides of the equation to eliminate the square root:

(x-1) = (3 - √(3x-2))^2

Expanding the right side of the equation:

(x-1) = (3 - √(3x-2))(3 - √(3x-2))

Using the FOIL method to multiply the two binomials:

(x-1) = 9 - 3√(3x-2) - 3√(3x-2) + (3x-2)

Simplifying:

x - 1 = 9 - 6√(3x-2) + 3x - 2

Combining like terms:

x - 1 = 3x + 7 - 6√(3x-2)

Rearranging the equation:

-2x - 8 = -6√(3x-2)

Dividing both sides by -2:

x + 4 = 3√(3x-2)

Cubing both sides of the equation to eliminate the cube root:

(x + 4)^3 = (3√(3x-2))^3

Expanding the left side of the equation:

x^3 + 12x^2 + 48x + 64 = 27(3x-2)

Simplifying:

x^3 + 12x^2 + 48x + 64 = 81x - 54

Rearranging the equation:

x^3 + 12x^2 + 48x - 81x + 64 + 54 = 0

Combining like terms:

x^3 + 12x^2 - 33x + 118 = 0

Unfortunately, this equation does not have a simple solution. It may be necessary to use numerical methods or a graphing calculator to find an approximate solution.