  The set of finitely generated subgroups of the group $PL_+(I)$ of
orientation-preserving piecewise-linear homeomorphisms of the unit interval
includes many important groups, most notably R.~Thompson's group $F$. In this
paper we show that every finitely generated subgroup $G<PL_+(I)$ is either
soluble, or contains an embedded copy of Brin's group $B$, a finitely
generated, non-soluble group, which verifies a conjecture of the first author
from 2009. In the case that $G$ is soluble, we show that the derived length of
$G$ is bounded above by the number of breakpoints of any finite set of
generators. We specify a set of `computable' subgroups of $PL_+(I)$ (which
includes R. Thompson's group $F$) and we give an algorithm which determines in
finite time whether or not any given finite subset $X$ of such a computable
group generates a soluble group. When the group is soluble, the algorithm also
determines the derived length of $\langle X\rangle$. Finally, we give a
solution of the membership problem for a family of finitely generated soluble
subgroups of any computable subgroup of $PL_+(I)$.
