28.9 Absolute Value Solution Verification

Absolute Value Solution Verification is the process of confirming that a solved absolute value equation or inequality genuinely satisfies the original relation exactly as it was first presented, drawing together the verification concepts already established for equations, inequalities, and compound inequalities into a single confirmatory process suited specifically to the dual nature of absolute value.

Equation Candidate Substitution is the action, applied when the original relation is an equation, of taking each value produced through Distinct Solution Collection and substituting it into the original equation exactly as first written, with the absolute value bars still intact, before evaluating anything further.

Both Equation Branches Checked is the requirement that this substitution be performed separately for every candidate value obtained, from both Independent Positive-Branch Resolution and Independent Negative-Branch Resolution, evaluating the enclosed expression at each candidate, applying the absolute value operation to that result, and confirming agreement with the original radius value for each candidate individually, since one branch's candidate producing a correct result offers no guarantee that the other branch's candidate does as well.

Inequality Boundary Evaluation is the action, applied when the original relation is an inequality, of substituting each of the two boundary values, identified through Absolute Value Center Identification combined with Positive Radius Identification, into the original inequality and confirming whether each boundary is correctly included or excluded, matching the strict or inclusive nature of the original comparison symbol.

Interior Sample Evaluation is the action, applied specifically to a solved Interior Absolute Value Inequalities case, of selecting a specific numerical value from within the Centered Bounded Interval claimed as the solution, substituting it into the original absolute value inequality, and confirming that it produces a true statement, mirroring Interior-region confirmation already familiar from Included-Side Sample Test.

Exterior Sample Evaluation is the corresponding action, applied specifically to a solved Exterior Absolute Value Inequalities case, of selecting a specific numerical value from within one of the two regions produced by Left Exterior Inequality Branch or Right Exterior Inequality Branch, substituting it into the original absolute value inequality, and confirming that it produces a true statement, while also selecting a value from the excluded middle region and confirming that it produces a false statement.

Interval and Number-Line Agreement is the action of comparing every representation produced for the solved absolute value relation, whether a symbolic compound inequality, an interval notation reflecting a Centered Bounded Interval or a Disjoint Ray Result, or a graphical number-line depiction, confirming that all such representations describe the identical set of values, in fulfillment of the same standard already established generally by Inequality Solution Representation Agreement and Compound Graph and Interval Agreement.

Original Absolute Relation Recheck is the concluding verification action, returning to the absolute value equation or inequality exactly as it was originally presented, before Preliminary Absolute Value Isolation, Linear Branch Equation Resolution, Interior Compound Resolution, or Exterior Branch Resolution were applied, and confirming that every stage of the solving process, from isolating the absolute value expression through splitting it into its dual cases and solving each resulting piece, faithfully preserves the meaning of that original statement.