Ixazomib Plus Pomalidomide and Dexamethasone in Treating Patients With Relapsed or Relapsed/Refractory Multiple Myeloma

Inclusion Criteria: * Voluntary written informed consent must be given before performance of any study related procedure not part of standard medical care, with the understanding that consent may be withdrawn by the patient at any time without prejudice to future medical care * Females of childbearing potential (FCBP) must have a negative serum or urine pregnancy test with a sensitivity of at least 25 mIU/mL within 10-14 days prior to and again within 24 hours of starting pomalidomide or MLN9708 and must either commit to continued abstinence from heterosexual intercourse or begin TWO acceptable methods of birth control, one highly effective method and one additional effective method AT THE SAME TIME, at least 28 days before she starts taking pomalidomide or MLN9708 through 90 days after the last dose of study drug; FCBP must also agree to ongoing pregnancy testing; men must agree to use a latex condom during sexual contact with a FCBP even if they have had a vasectomy from the time of signing the informed consent form through 90 days after the last dose of study drug; all patients must be counseled at a minimum of every 28 days about pregnancy precautions and risks of fetal exposure * All patients enrolled into this trial, must be registered in and must comply with all requirements of the POMALYST (pomalidomide) Risk Evaluation and Mitigation Strategy (REMS) program * Patients must have a diagnosis of relapsed or relapsed and refractory multiple myeloma with a minimum of one prior regimen and a maximum of 5 prior regimens * Patients must have had therapy with a proteasome inhibitor and lenalidomide and be refractory to lenalidomide according to the International Myeloma Working Group (IMWG) criteria definition of refractory disease (progressive disease on or within 60 days of stopping lenalidomide) * Patients must have measurable disease defined as one of the following: * Serum M protein \>= 0.5 g/dL * Urine M protein \>= 200 mg/24 hours * Serum free light chain \>= 10 mg/dL provided the free light chain (FLC) ratio is abnormal * Eastern Cooperative Oncology Group (ECOG) performance status 0, 1, or 2 * Absolute neutrophil count (ANC) \>= 1,000/mm\^3 * Platelet count \>= 75,000/uL for patients in whom \< 50% of bone marrow nucleated cells are plasma cells; or a platelet count \>= 50,000/uL for patients in whom \>= 50% of bone marrow nucleated cells are plasma cells; platelet transfusions are not allowed within 3 days of last platelet assessment to confirm eligibility * Total bilirubin =\< 1.5 × the institutional upper limit of the normal range (IULN) * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 3 × IULN * Calculated creatinine clearance \>= 45 mL/min Exclusion Criteria: * Female patients who are pregnant or breastfeeding or have a positive serum pregnancy test during the screening period * Failure to have fully recovered (ie, =\< grade 1 toxicity) from the reversible effects of prior chemotherapy * Prior therapy with a combination regimen containing pomalidomide except the 2 drug combination of pomalidomide and dexamethasone * Major surgery within 14 days before enrollment * Radiotherapy within 14 days before enrollment; if the involved field is small, 7 days will be considered a sufficient interval between treatment and administration of the MLN9708 * Central nervous system involvement * Infection requiring systemic antibiotic therapy or other serious infection within 14 days before study enrollment * Evidence of current uncontrolled cardiovascular conditions, including uncontrolled hypertension, uncontrolled cardiac arrhythmias, symptomatic congestive heart failure, unstable angina, or myocardial infarction within the past 6 months * Systemic treatment, within 14 days before the first dose of MLN9708, with strong inhibitors of cytochrome P450, family 1, subfamily A, polypeptide 2 (CYP1A2) (fluvoxamine, enoxacin, ciprofloxacin), strong inhibitors of cytochrome P450, family 3, subfamily A (CYP3A) (clarithromycin, telithromycin, itraconazole, voriconazole, ketoconazole, nefazodone, posaconazole) or strong CYP3A inducers (rifampin, rifapentine, rifabutin, carbamazepine, phenytoin, phenobarbital), or use of Ginkgo biloba or St. John's wort * Unable or unwilling to undergo antithrombotic prophylaxis * Ongoing or active systemic infection, active hepatitis B or C virus infection, or known human immunodeficiency virus (HIV) positive * Any serious medical or psychiatric illness that could, in the investigator's opinion, potentially interfere with the completion of treatment according to this protocol * Known allergy to any of the study medications, their analogues, or excipients in the various formulations of any agent * Known gastrointestinal (GI) disease or GI procedure that could interfere with the oral absorption or tolerance of MLN9708 or pomalidomide including difficulty swallowing * Diagnosed or treated for another malignancy within 2 years before study enrollment or previously diagnosed with another malignancy with evidence of residual disease; patients with nonmelanoma skin cancer or carcinoma in situ of any type are not excluded if they have undergone complete resection * Patient has \> grade 2 peripheral neuropathy on clinical examination during the screening period * Participation in other clinical trials, including those with other investigational agents not included in this trial, within 21 days of the start of this trial and throughout the duration of this trial (for all other standard therapies, no treatment within 14 days of the start of this trial) * Patients who are pomalidomide refractory, defined as patients who progress on or within 60 days of pomalidomide when given as a single agent or with dexamethasone