Ultrathin transition metal dichalcogenides (TMDCs) have recently been extensively investigated to understand their electronic and optical properties. Here we study ultrathin Mo0.91W0.09Te2, a semiconducting alloy of MoTe2, using Raman, photoluminescence (PL), and optical absorption spectroscopy. Mo0.91W0.09Te2 transitions from an indirect to a direct optical band gap in the limit of monolayer thickness, exhibiting an optical gap of 1.10 eV, very close to its MoTe2 counterpart. We apply tensile strain, for the first time, to monolayer MoTe2 and Mo0.91W0.09Te2 to tune the band structure of these materials; we observe that their optical band gaps decrease by 70 meV at 2.3% uniaxial strain. The spectral widths of the PL peaks decrease with increasing strain, which we attribute to weaker exciton–phonon intervalley scattering. Furthermore, strained MoTe2 and Mo0.91W0.09Te2 extend the range of band gaps of TMDC monolayers further into the near-infrared, an important attribute for potential applications in optoelectronics.