Chapter 25: Graphing Inequalities Containing Absolute Values: Situation 19 From the MACMTL–CPTM Situations Project

This episode occurred during a course for prospective secondary mathematics teachers. The discussion focused on the graph of y – 2 ≤ |x + 4|. The instructor demonstrated how to graph this inequality using compositions of transformations, generating the graph in Figure 25.1. Students proposed other methods, which included the two different symbolic formulations and accompanying graphs as seen in Figure 25.2 and Figure 25.3. Students expected their graphs to match the instructor's graph, and they were confused by the differences they noticed.

This Prompt involves the graph of an absolute value inequality involving two variables. The graph of an absolute value inequality is related to the graph of an absolute value function. Absolute values can be interpreted on the basis of a symbolically stated definition of absolute value as well as based on the conception of absolute value as distance from 0. Composite functions such as absolute value functions can be viewed as transformations of input and/or output values of functions. Working with absolute values entails keeping track of the logic of the conjunctions and disjunctions into which absolute value statements transform, and working with inequalities involves applying an appropriate range of rules governing their transformation. Inequalities in two variables can be interpreted by examining the inequality for each of a sequence of constant values for one of the variables.