The rapid proliferation of antibiotic-resistant pathogens has spurred the use of drug combinations to maintain clinical efficacy and combat the evolution of resistance. Drug pairs can interact synergistically or antagonistically, yielding inhibitory effects larger or smaller than expected from the drugs' individual potencies. Clinical strategies often favor synergistic interactions because they maximize the rate at which the infection is cleared from an individual, but it is unclear how such interactions affect the evolution of multi-drug resistance. We used a mathematical model of in vivo infection dynamics to determine the optimal treatment strategy for preventing the evolution of multi-drug resistance. We found that synergy has two conflicting effects: it clears the infection faster and thereby decreases the time during which resistant mutants can arise, but increases the selective advantage of these mutants over wild-type cells. When competition for resources is weak, the former effect is dominant and greater synergy more effectively prevents multi-drug resistance. However, under conditions of strong resource competition, a tradeoff emerges in which greater synergy increases the rate of infection clearance, but also increases the risk of multi-drug resistance. This tradeoff breaks down at a critical level of drug interaction, above which greater synergy has no effect on infection clearance, but still increases the risk of multi-drug resistance. These results suggest that the optimal strategy for suppressing multi-drug resistance is not always to maximize synergy, and that in some cases drug antagonism, despite its weaker efficacy, may better suppress the evolution of multi-drug resistance.