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.