This study investigated preservice secondary mathematics teachers' difficulties with and understanding of graphs of two-variable inequalities with the lens of the APOS theory. The analysis showed that most teachers understood inequality graphs as actions, at best, and used informal rules to construct graphs. They also had difficulty justifying graphs, such as overgeneralization of linear inequality graphs, lack of understanding of the meaning of solutions and the Cartesian Connection, and inability to transfer algebraic/verbal to geometric rays. This study suggests two methods for the visualization of inequality graphs, which uses the concepts of variable and parameter. [For the complete proceedings, see ED606556.]