This paper describes the behavior of charged, polymer brushes in electrolyte solutions of varying solvent quality. The brush height, d, dependence on the chain length, L (=Nl, where I is the Kuhn length), the grafting density sigma, and solvent conditions is determined. We consider a monomer-monomer potential consisting of three components : (1) a long-ranged, screened Coulombic component of strength (v) over bar/l (1 is the Kuhn length) and range kappa(-1); (2) a short-ranged, two-body component of strength <(w)over bar l>; and (3) a short-ranged, three-body component of strength (u) over bar l(3). In particular, we examine the transition from a stretched state to a collapsed state in a poor solvent; (<(w)over bar <0>) as the solvent quality is decreased. Using dimensional analysis, Monte Carlo methods, and a variational technique, a first order transition is observed as predicted by the scaling arguments of Ross et al. and Borisov et al. for high charge/grafting densities. Using a variational procedure, we derive an analytical expression for the brush size and determine, quantitatively, the critical conditions for a first order transition in terms of key dimensionless variables, vN(5/2), kappa lN(1/2), wN(3/2),and uN(2) (where v = 2 pi sigma l(2) (v) over bar, w = sigma l(2) (w) over bar, and u = sigma(2)l(4) (u) over bar.