We consider a few-boson system confined to one dimension with a single distinguishable particle of lesser mass. All particle interactions are modeled with δ functions, but due to the mass imbalance the problem is nonintegrable. Universal few-body binding energies, atom-dimer and atom-trimer scattering lengths, are all calculated in terms of two parameters, namely the mass ratio mL/mH, and ratio gHH/gHL of the δ-function couplings. We specifically identify the values of these ratios for which the atom-dimer or atom-trimer scattering lengths vanish or diverge. We identify regions in this parameter space in which various few-body inelastic processes become energetically allowed. In the Tonks-Girardeau limit (gHH→∞), our results are relevant to experiments involving trapped fermions with an impurity atom.