Ixazomib Citrate in Treating Patients With Relapsed Multiple Myeloma That Is Not Refractory to Bortezomib

Inclusion Criteria: * Calculated creatinine clearance (using Cockcroft-Gault equation) \>= 30 mL/min (obtained =\< 14 days prior to registration) * Absolute neutrophil count \>= 1000/mL (obtained =\< 14 days prior to registration) * Untransfused platelet count \>= 75000/mL (obtained =\< 14 days prior to registration) * Hemoglobin \>= 8.0 g/dL (obtained =\< 14 days prior to registration) * Total bilirubin =\< 1.5 x the upper limit of the normal range (ULN) (obtained =\< 14 days prior to registration) * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 3 x ULN (obtained =\< 14 days prior to registration) * Patients with relapsed multiple myeloma who have already received one or more standard treatment regimens * Measurable disease of multiple myeloma as defined by at least ONE of the following: * Serum monoclonal protein \>= 1.0 g/dL * \>= 200 mg of monoclonal protein in the urine on 24 hour electrophoresis * Serum immunoglobulin free light chain \>= 10 mg/dL AND abnormal serum immunoglobulin kappa to lambda free light chain ratio * For patients with extramedullary disease (EMD) measurable disease by computed tomography (CT) or magnetic resonance imaging (MRI) or the CT portion of the positron emission tomography (PET)/CT: Must have at least one lesion that has a single diameter of \>= 2 cm. Skin lesions can be used if the area is \>= 2 cm in at least one diameter and measured with a ruler * Plasma cell count \>= 0.5 x 10\^9/L or 5 percent of t…