This phase I trial studies the side effects and the best dose of trametinib when given together with combination chemotherapy and radiation therapy in treating patients with stage III non-small cell lung cancer that cannot be removed by surgery. Trametinib may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. Drugs used in chemotherapy, such as carboplatin and paclitaxel, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Radiation therapy uses high energy x-rays to kill tumor cells. Giving trametinib, combination chemotherapy, and radiation therapy may be a better treatment for non-small cell lung cancer.