Two–weight criteria of various type for one–sided maximal functions and one–sided potentials are established in variable exponent Lebesgue spaces. Among other re- sults we derive the Hardy–Littlewood, Fefferman–Stein and trace inequalities in these spaces. Weighted estimates for Hardy–type, maximal, potential and singular opera- tors defined by means of a quasi–metric and a doubling measure are derived in Lp(x) spaces. In some cases examples of weights guaranteeing the appropriate weighted estimates are given.