Inequivalent Equations: Visual Checks for Algebraic Transformations

This GeoGebra lesson demonstrates how seemingly reasonable algebraic transformations can produce inequivalent equations, leading to different solution sets. It highlights the key idea of equivalence vs. inequivalence in equation solving and the corresponding interpretation of solution sets when restrictions or operations change the validity of steps. Core topics include inequality/inequivalent equations, criteria for when a transformation preserves solutions, and the use of geometric intuition—such as the geometric mean and relationships involving secant lines—to make the “solution preservation” (or loss) visible. The provided construction summary is empty, so no specific GeoGebra commands/objects can be listed; the construction is best understood as using a geometric/algebraic representation to test candidate solutions and compare solution sets before and after the transformation. Suitable for secondary school through early undergraduate learners, especially for lessons on algebraic reasoning and misconception correction.