We develop a revealed preference method for studying preferences of individuals whose choices are inconsistent with utility maximisation. We investigate a condition that imposes acyclicity on a notion of strict preferences induced by the observable data, but not on the weak ones. We show that this restriction is necessary and sufficient for the data to be rationalisable with approximate utility maximisation, where an alternative is selected from a menu only if its utility is not significantly lower than that of any other available option. This provides a test for consumer behaviour that may be subject to imprecision, vagueness of judgement, or incommensurability, among others. Moreover, although the individual may fail to maximise their utility exactly, it may be possible to recover their true preferences from observations. Our results require minimal assumptions on the empirical framework and are applicable, amongst others, to the study of choices over consumption bundles, state-contingent consumption, and lotteries.