26.7 Compound Inequality Error Analysis

Compound Inequality Error Analysis is the systematic identification and correction of mistakes that occur while solving conjunctive or disjunctive compound inequalities, cataloguing the specific ways a solving attempt can go wrong beyond the errors already possible when solving a single, uncombined linear inequality.

And-Or Connector Confusion occurs when a solver mistakes a compound inequality joined by "and" for one joined by "or," or the reverse, applying Simultaneous Truth Requirement where At-Least-One Truth Requirement was called for, or applying the reverse, resulting in a solution method fundamentally mismatched to the actual connector present in the original statement.

Intersection and Union Confusion occurs when a solver correctly identifies the connector but nonetheless combines the two individual solution sets incorrectly, applying Combined Outer Solution Regions when Overlapping Solution Selection was required for a conjunctive inequality, or applying the reverse when a disjunctive inequality was intended, producing a final solution set either too broad or too narrow relative to the true answer.

Partial Three-Part Transformation occurs during Chained Inequality Resolution when an additive or multiplicative operation is applied to only two of the three parts of the chain, such as the left and middle parts, while the right part is left untransformed, violating Same Additive Change across Three Parts or Same Positive Scale across Three Parts and producing a chain no longer equivalent to the original.

Single-Symbol Reversal after Negative Scaling occurs during Chained Inequality Resolution when a negative quantity is used to scale all three parts of a chain, but only one of the two comparison symbols within the chain is reversed rather than both, violating Reversal of Both Comparison Symbols and leaving the chain internally inconsistent between its two halves.

Incorrect Compound Endpoint Inclusion occurs when the inclusion or exclusion of one of the two boundary values in a compound inequality is assigned incorrectly, mismatching Open and Closed Compound Endpoints against the actual strict or inclusive nature of the comparison that produced that boundary, whether in a chained form or a separated form.

Missing One Comparison Branch occurs when a solver, while carrying out Independent Comparison Resolution or Independent Disjunctive Resolution, solves only one of the two individual inequalities in a Separated Comparison Form and neglects the second entirely, producing a final solution set based on only half of the original compound statement.

Empty Conjunction Misclassification occurs when a solver fails to recognize that the two individual solution sets of a conjunctive inequality do not overlap at all, mistakenly reporting some nonempty region as the solution rather than correctly identifying an Empty Shared Region through accurate Conjunctive Boundary Comparison.

All-Real Disjunction Misclassification occurs when a solver fails to recognize that the two individual solution sets of a disjunctive inequality, once combined, actually cover every real number without gap, mistakenly reporting a Disjoint Ray Result or a partial union rather than correctly identifying the All-Real Disjunctive Result that Overlapping Disjunctive Regions actually produces.

Compound Inequality Correction is the concluding action of the analysis, in which the specific error identified among the categories above is traced to its exact originating step, whether in connector identification, three-part transformation, endpoint assignment, individual branch resolution, or final combination, and that step is redone correctly. Correction concludes only once the corrected solution passes Compound Solution Verification, including a successful Connector Meaning Check, correctly assigned boundary tests, and confirmed Compound Graph and Interval Agreement across every representation of the result.