  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 $\dot{M}_B$.
