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<p/> <p>Two-weighted norm estimates with general weights for Hardy-type transforms and potentials in variable exponent Lebesgue spaces defined on quasimetric measure spaces <inline-formula> <graphic file="1029-242X-2010-329571-i1.gif"/></inline-formula> are established. In particular, we derive integral-type easily verifiable sufficient conditions governing two-weight inequalities for these operators. If exponents of Lebesgue spaces are constants, then most of the derived conditions are simultaneously necessary and sufficient for corresponding inequalities. Appropriate examples of weights are also given.</p>