The marked deviation from Fermi liquid behavior for the quasi-one-dimensional (1D) purple bronze Li_0.9Mo₆O₁₇ (LMO) has been observed both by bulk transport and by surface-sensitive spectroscopic probes, and has generated much theoretical interest. Here we present spectroscopic scanning tunneling microscope (STM) images of 1D 'chains' on the surface of LMO. We find that the Coulomb suppression of tunneling around the Fermi energy is inhomogenous on the nanometer length scale, which necessitates a revision of the existing model of Luttinger-liquid-like behavior that was based on spatially-averaged spectroscopic techniques. We further report that chains of varying length exhibit resonances at varying energies, and we discuss the implications of this phenomenon for modeling the surface structure of LMO as a 1D quantum system.