25.6 Solved Inequality Representation

Solved Inequality Representation is the concluding stage of solving a one-variable linear inequality, in which the One-Sided Solution Inequality produced by the solving sequence is verified for correctness and then expressed in whichever notation the situation calls for, confirming both the boundary value and the direction of the inequality before the solution is finalized.

Isolated Inequality Statement is the direct output of the solving process, presenting the variable alone on one side of an inequality symbol, compared to a single numerical boundary value, matching the Single-Boundary Solution Form required within this scope and serving as the starting point for every subsequent verification and representation action.

Solved Boundary Inclusion Check is the action of confirming whether the Isolated Inequality Statement uses a Strict Comparison or an Inclusive Comparison, reviewing the symbol carried through every Additive Inequality Transformation and Multiplicative Inequality Transformation applied during solving, including any Relation Reversal after Negative Scaling, to ensure that the inclusive or exclusive nature of the original inequality has been preserved correctly into the final statement.

Included-Side Sample Test is a verification action performed by selecting a specific numerical value from the side of the boundary that the Isolated Inequality Statement claims satisfies the inequality, substituting that value into the original, unsolved inequality, and confirming that it produces a true statement, providing direct evidence that the solving process correctly identified which side of the boundary belongs to the solution set.

Excluded-Side Sample Test is the complementary verification action, selecting a specific numerical value from the side of the boundary that the Isolated Inequality Statement claims does not satisfy the inequality, substituting that value into the original, unsolved inequality, and confirming that it produces a false statement, providing evidence from the opposite direction that the boundary and its associated direction were identified correctly.

Original Inequality Substitution is the shared technique underlying both Included-Side Sample Test and Excluded-Side Sample Test: the chosen test value is always substituted into the inequality exactly as it was first presented, before any reduction, distribution, or scaling was applied, ensuring that the verification tests the true original relationship rather than an intermediate or possibly flawed transformed version.

Final Symbol-Interval-Ray Alignment is the concluding action of this stage, in which the confirmed Isolated Inequality Statement is translated into whichever of the established notations the context requires, whether Symbolic Inequality Form, One-Sided Interval Form, or a graphical Number-Line Solution Representation, with each element of the chosen notation, including boundary markers, enclosing symbols, and ray direction, checked for agreement with the confirmed inequality symbol following Boundary Marker and Symbol Agreement and Inequality Solution Representation Agreement.