This paper is devoted to systems of n inhomogeneous equations of hydrodynamic type with two independent variables. Using a geometric formalism for such systems which goes back to Riemann, one can associate with every system of hydrodynamic type a vector field and a field of linear operators acting on an appropriate tangent bundle. In terms of these fields, we obtain a number of tests for inhomogeneous systems of hydrodynamic type to be decoupled into subsystems with block-triangular interaction. These tests supplement Bogoyavlenskii's well-known results on decoupling of homogeneous systems of hydrodynamic type into non-interacting subsystems.