In this paper, a new family of distributions called exponentiated TX distribution is defined. Some of its properties and special cases are discussed. A member of the family, namely, the three-parameter exponentiated Weibullexponential distribution is defined and studied. Some of its properties including distribution shapes, limit behavior, hazard function, Shannon entropy, moments, skewness and kurtosis are discussed. The flexibility of the exponentiated Weibull-exponential distribution is assessed by applying it to three real data sets and comparing it with other distributions. The exponentiated Weibull-exponential distribution is found to adequately fit left-skewed and right-skewed data sets.