This study utilizes the Cartesian Connection and the action, process, object, and schema (APOS) theory to investigate how preservice mathematics teachers constructed and justified graphs of two-variable inequalities. The analysis shows that most teachers were not equipped with the skills and knowledge that were prerequisites for action, process, and object conceptions of two-variable inequality graphs. Many teachers used informal rules, rather than mathematical ideas such as the Cartesian Connection, to construct and/or justify inequality graphs. They also perceived inequality graphs as shapes rather than as representations based on the meaning of inequalities. I present a genetic decomposition for inequality graphs, through the concepts of variable and of parameter, as well as tasks associated with the genetic decomposition. I call for more studies on inequalities, including those relating to the implementation of the suggested genetic decomposition and tasks.