This randomized phase I/II trial studies the side effects and best dose of pomalidomide and ixazomib when given together with dexamethasone and to see how well pomalidomide and dexamethasone with or without ixazomib works in treating patients with refractory multiple myeloma. Biological therapies, such as pomalidomide and dexamethasone, may stimulate the immune system in different ways and stop tumor cells from growing. Ixazomib may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. It is not yet known whether pomalidomide and dexamethasone are more effective with or without ixazomib in treating multiple myeloma.