A measurement of the underlying activity in scattering processes with a pT scale in the several GeV region is performed in proton-proton collisions at sqrt(s) = 7 TeV, using data collected by the CMS experiment at the LHC. The production of charged particles with pseudorapidity eta < 2 and pT > 0.5 GeV/c is studied in the azimuthal region transverse to that of the leading set of charged particles forming a “track-jet”. A significant growth of the average multiplicity and scalar pT sum of those particles is observed with increasing pT of the leading track-jet, followed by saturation above a few GeV/c. A hardening of the multiplicity distribution, of the scalar pT sum distribution, and of the charged particle pT spectrum is also observed. For track-jet pT larger than a few GeV/c, the activity in the transverse region is approximately doubled with a centre-of-mass energy increase from 0.9 to 7 TeV. Predictions of several QCD-inspired models as implemented in PYTHIA are compared to the data.