  We consider a few-boson system confined to one dimension with a single
distinguishable particle of lesser mass. All particle interactions are modeled
with $\delta$-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: $m_{\text{L}}/m_{\text{H}}$, and ratio
$g_{\text{HH}}/g_{\text{HL}}$ of the $\delta$-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 process become
energetically allowed. In the Tonks-Girardeau limit ($g_{\text{HH}}\rightarrow
\infty$), our results are relevant to experiments involving trapped fermions
with an impurity atom.
