The purpose of this study was to investigate the nature of preservice mathematics teachers' understanding of inequalities. The design of the study was a multi-case study of secondary mathematics education preservice teachers with a focus on their understanding and the lessons that they plan and implement. Data from individual interviews, observations, and documents were collected and analyzed. Five preservice mathematics teachers from a major university in the Southeast United States participated in this study. These preservice teachers were examined during their student teaching field experience semester. Preservice teachers displayed a robust procedural understanding, which involved an ability to apply the procedures necessary for solving inequalities, during pre-student teaching interviews (Rittle-Johnson & Alibali, 1999). At times, their understanding extended beyond an operational view of inequalities and allowed them to consider the relational nature of inequalities (Knuth et al., 2006). This was especially true while discussing the meaning of a solution of an inequality. Preservice teachers displayed some of the same conceptions noted within prior literature (e.g., treating inequalities as equations, believing that solutions of inequalities must be inequalities). There did not appear to be a relationship between a preservice teacher displaying one conception and any other conception. Preservice teachers' understanding of inequalities played a role in their planning and implementation of lessons related to systems of linear inequalities and/or quadratic inequalities. Boundary lines or curves of inequalities seemed to be a recurring focal point where preservice teachers' understanding influenced their lessons. Preservice teachers often used and explained the graphing of boundary lines or curves as an extension of graphing corresponding equations; thereby providing opportunities for students to review prior skills and concepts with linear and quadratic equations. Solution strategies were presented in an operational manner which tended to treat boundary lines or curves as an isolated component. This isolation often carried over to descriptions of solutions of inequalities. Preservice teachers described solutions of inequalities as regions and did not mention the inclusion or exclusion of boundary lines or curves. Overall, preservice teachers seemed to view and treat inequalities as something to do rather than something to be interpreted. Implications of this study indicate a need to consider changes in methods courses. Deficiencies in preservice teachers' understanding of inequalities need to be addressed within methods courses. Preservice teachers should have opportunities to examine pedagogical decisions that foster a relational view of inequalities (and equations). Additionally, methods courses should help develop preservice teachers' awareness of how technology can be integrated into lessons without reducing tasks to rote button pushing exercises. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com.bibliotheek.ehb.be/en-US/products/dissertations/individuals.shtml.]