If there exist a solution satifying the inequality constraint, then I would recommend using Newton-Raphson, starting with an initial value that satisfies the constraint. If that inital value is well within the constraint area, Newton-Raphson should converge to a solution in that area but the boundary of such soutions can be very complex- even fractal.
If there does not exist such a solution, you would have to try to find a value that satisfies the constraint and comes "as close as possible" to satisfying the equation- Prove It's "optimization".
Solve the unconstrained system using Newton-Raphson then either the constraints are satisfied at the solution in which case you are done, or not. If not there are either multiple solutions to the unconstrained problem or no solution. If there are multiple solutions you need to restart N-R at a different point outside of the basin of convergence to your previous solution/s and see where that takes you untill you find a solution satisfying the constraints.
Looking at this problem I suspect you may well have started with a constrained minimisation(maximisation) problem which you differentiated and set the derivatives to zero. If this is the case this is a . You should look at constrained optimisation algorithms.
CB