  The density dependence of the nuclear symmetry energy governs important
aspects of very neutron rich systems such as heavy nuclei and their collisions,
neutron stars and their mergers. Many analyses of experimental data have
generated constraints on the symmetry energy and its first derivative at
saturation density, rho_0 ~ 2.7xE14 g/cm3.
  We show that each analysis does not accurately constrain the symmetry energy
at rho_0, but rather at a lower density, rho_s, that is most sensitively probed
by that analysis. Using published constraints on the symmetry energy and its
first derivative at rho_0, we constrain the symmetry energy within a density
range of 0.25rho_0 to 0.75rho_0 that is relevant to the inner crusts of neutron
stars and to the neutrino-sphere of core-collapse supernovae. With appropriate
data, the symmetry energy can be similarly constrained over a wider density
range.
