The classic Bondi solution remains a common starting point both for studying black hole growth across cosmic time in cosmological simulations and for smaller scale simulations of AGN feedback. In nature, however, there will be inhomogenous distributions of rotational velocity and density along the outer radius ($R_o$) marking the sphere of influence of a black hole... While there have been many studies of how the Bondi solution changes with a prescribed angular momentum boundary condition, they have all assumed a constant density at $R_o$. In this Letter, we show that a non-uniform density at $R_o$ causes a meridional flow and due to conservation of angular momentum, the Bondi solution qualitatively changes into an inflow-outflow solution. Using physical arguments, we analytically identify the critical logarithmic density gradient $|\partial{\ln{\rho}}/\partial{\theta}|$ above which this change of the solution occurs. For realistic $R_o$, this critical gradient is less than 0.01 and tends to 0 as $R_o \rightarrow \infty$. We show using numerical simulations that, unlike for solutions with an imposed rotational velocity, the accretion rate for solutions under an inhomogenous density boundary condition remains constant at nearly the Bondi rate $\dot{M}_B$, while the outflow rate can greatly exceed $\dot{M}_B$. (read more)