  In this paper we consider Iterated Function Systems (IFS) on the real line
consisting of continuous piecewise linear functions. We assume some bounds on
the contraction ratios of the functions, but we do not assume any separation
condition. Moreover, we do not require that the functions of the IFS are
injective, but we assume that their derivatives are separated from zero. We
prove that if we fix all the slopes but perturb all other parameters, then for
all but a set of less than full packing dimensional exceptional set, the
Hausdorff dimension of the attractor is equal to the exponent which comes from
the most natural system of covers of the attractor.
