To solve the equation x + 1/3 = 3/x - 1, we can start by getting rid of the fractions by multiplying every term by the least common denominator, which is 3x:

3x(x + 1/3) = 3x(3/x - 1)

Expanding both sides:

3x^2 + x = 9 - 3x

Rearranging the equation:

3x^2 + x + 3x - 9 = 0

Combining like terms:

3x^2 + 4x - 9 = 0

Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 3, b = 4, and c = -9. Substituting these values into the quadratic formula:

x = (-4 ± √(4^2 - 4(3)(-9))) / (2(3))

Simplifying:

x = (-4 ± √(16 + 108)) / 6

x = (-4 ± √124) / 6

x = (-4 ± 2√31) / 6

Simplifying further:

x = (-2 ± √31) / 3

Therefore, the solutions to the equation x + 1/3 = 3/x - 1 are x = (-2 + √31) / 3 and x = (-2 - √31) / 3.