Since the seminal PPAD-completeness result for computing a Nash equilibrium even in two-player games, an important line of research has focused on relaxations achievable in polynomial time. In this paper, we consider the notion of $\varepsilon$-well-supported Nash equilibrium, where $\varepsilon \in [0,1]$ corresponds to the approximation guarantee. Put simply, in an $\varepsilon$-well-supported equilibrium, every player chooses with positive probability actions that are within $\varepsilon$ of the maximum achievable payoff, against the other player's strategy. Ever since the initial approximation guarantee of 2/3 for well-supported equilibria, which was established more than a decade ago, the progress on this problem has been extremely slow and incremental. Notably, the small improvements to 0.6608, and finally to 0.6528, were achieved by algorithms of growing complexity. Our main result is a simple and intuitive algorithm, that improves the approximation guarantee to 1/2. Our algorithm is based on linear programming and in particular on exploiting suitably defined zero-sum games that arise from the payoff matrices of the two players. As a byproduct, we show how to achieve the same approximation guarantee in a query-efficient way.