Motivated by the magnetic properties of the spin-chain compounds LiCuSbO4≡LiSbCuO4 and Rb2Cu2Mo3O12, we study the ground state of the Heisenberg chain with dimerized nearest-neighbor ferromagnetic (FM) (J1,J1′<0) and next-nearest-neighbor antiferromagnetic (J2>0) couplings. Using the density-matrix renormalization group technique and spin-wave theory, we find a first-order transition between a fully polarized FM and an incommensurate spiral state at 2α=β/(1+β), where α is the frustration ratio J2/|J1| and β the degree of dimerization J1′/J1. In the singlet spiral state the spin-gap is vanishingly small in the vicinity of the FM transition, corresponding to a situation of LiCuSbO4. For larger α, corresponding to Rb2Cu2Mo3O12, and smaller β there is a crossover from this frustration induced incommensurate state to an Affleck-Lieb-Kennedy-Tasaki-type valence-bond solid state with substantial spin gaps.