Few objective constraints exist on radial density variations in planetary interiors. Even if the mean density and mean moment of inertia were known with perfect accuracy, they would only provide integral constraints on the density models. However, among the family of monotonic radial density models with a given mean density and mean moment of inertia, the simple two-layer piecewise constant models have useful extremal properties. The model has only three parameters, an inner density, an outer density, and a transition radius. Once the radial moment constraints are applied, there is only one remaining degree of freedom. If the outer region density is somehow specified, then the inner region density of the two-layer model provides a firm lower bound on central density of monotonic models. Likewise, if an upper bound on central density can be provided, then the two-layer model provides a lower bound on outer region density. An envelope of acceptable density models can be generated by scanning the transition depth from the center to the surface. Any monotonic model with specified mean density and mean inertial moment must lie within that envelope. Resulting extremal density envelopes for the Earth, Moon, and Mars are compared to published radial density profiles. For the Moon, the moment constraints are quite restrictive. For the Earth, which is more centrally condensed, the allowed envelope of density profiles is rather broad. Mars is an intermediate case. Present geodetic and astrometric observations only constrain the Martian mean moment of inertia to lie somewhere in the range 0.325≤I/MR2≤0.365. However, our analysis shows that any value within that range can be accommodated without invoking geochemically implausible density minima or maxima.