How to use the Karush–Kuhn–Tucker conditions?

Question

From what I read, the Karush-Kuhn-Tucker conditions are a generalization of the Lagrange Multiplier Method.

For the Lagrange Multiplier Method I have been able to find a serie of steps I must do to find the result, but I don't see quite clearly what I am supposed to do with the KKT conditions.

Lagrange Multiplier method:

1. Set the constraint function equal to 0
2. Create another function called the Lagrangian function by combining the objective function and the constraint
3. Take the partial derivatives of the Lagrangian and set them to 0
4. Solve the system of equations to get the values that minimize the objective function

KKT conditions method:

1. Define the Lagrangian function
2. Write the additionnal conditions
3. ??

What should I do? Do I need to solve a system of equations as before ? Do I need to check that the condition hold ? What if they don't hold ?