Sitting on the Fence: Geometric Solving of Absolute-Value Inequalities

This example demonstrates—via interactive geometry—the solution of absolute-value inequalities by visualizing a distance-based condition and identifying where it holds. It centers on key ideas such as rewriting absolute-value inequalities as equivalent statements about distances to fixed points, understanding boundary cases where the inequality switches, and using geometric loci to represent the solution set. In the GeoGebra construction, commands like Ray, Point, Mirror, and Vector help build the symmetric/reflective setup; Distance and Segment measure the relevant distances (e.g., |x−a| and distances to endpoints); Circle provides the boundary defined by a constant distance; Intersect determines critical intersection points; Polygon highlights the final admissible region. Additional commands such as Translate, Dilate, and Text are used to position transformed elements and label thresholds clearly. Suitable for middle/high school through early university learners, especially for teaching and learning the graphical/geometric method for absolute-value inequalities.