Various efficient matrix inequalities have recently been proposed to deal with the stability analysis of linear systems with time-varying delays. This paper provides more insights on the relationship between some of them. We present an equivalent formulation of Moon et al.'s inequality, allowing us to discover strong links not only with the most recent and efficient matrix inequalities such as the reciprocally convex combination lemma and also its relaxed version but also with some previous inequalities such as the approximation inequality introduced in [14] or free-matrix-based inequality. More especially, it is proved that these existing inequalities can be captured as particular cases of Moon et al.'s inequality. Examples show the best tradeoff between the reduction of conservatism and the numerical complexity.