A novel and general method for numerical integration of ODEs or DAEs with inequality path constraints involving state variables is proposed. In general, active inequality path constraints involving state variables produce high-index DAEs, which complicate the solution of dynamic simulation and optimization problems. Current DAE integrators can handle only certain limited classes of high-index problems. However, the recently developed method of dummy derivatives can be used to derive an index-1 problem with the same solution set as the high-index problem. This permits the use of standard DAE codes for integrating high-index DAEs, although the equivalent index-1 DAE has a larger number of equations than the original DAE system. Our method detects the activation and deactivation of inequality path constraints during integration, and solves the resulting high-index DAE system as necessary. In addition to allowing the solution of dynamic simulation problems with inequality path constraints, we show that this method simplifies the solution of dynamic optimization problems using the control parameterization method. The method allows us to handle the inequality path constraints involving state variables within the DAE integrator, resulting in fewer objective function and gradient evaluations for the NLP solver, and reducing the time necessary to solve the dynamic optimization problem.