26.5 Disjunctive Inequalities

Disjunctive Inequalities are compound inequalities in which a value belongs to the solution set as long as it satisfies at least one of the two individual comparisons, corresponding to the Disjunctive Compound Form introduced within Compound Inequality Scope and typically producing a solution set that extends outward in two separate directions rather than being confined between two boundaries.

At-Least-One Truth Requirement is the defining condition of this category: a candidate value must make at least one of the two joined comparisons true, though it need not satisfy both, in order to belong to the solution set, standing in direct contrast to the Simultaneous Truth Requirement governing conjunctive inequalities.

Independent Disjunctive Resolution is the opening action of solving a disjunctive inequality, applying whichever standard solving technique is appropriate to each of the two individual inequalities entirely on its own, exactly as in Independent Comparison Resolution for the conjunctive case, producing two separate instances of an Isolated Inequality Statement before any combination of the two results is attempted.

Combined Outer Solution Regions is the action of joining together the two individual solution sets produced by Independent Disjunctive Resolution, retaining every value that belongs to either one, in fulfillment of At-Least-One Truth Requirement, rather than restricting the result to only the values shared by both as Overlapping Solution Selection would for a conjunctive inequality.

Disjoint Ray Result describes the common outcome of Combined Outer Solution Regions when the two individual solution rays point away from each other with no shared values between them, such as one ray extending below a lower boundary and the other extending above a separate, higher boundary, producing a solution set consisting of two separate, unconnected regions rather than a single continuous span.

Overlapping Disjunctive Regions describes the alternative outcome in which the two individual solution rays share some values in common, such as when both rays extend in overlapping directions or one ray's region entirely contains the other's, in which case the combined solution set may simplify to a single continuous region, or in some cases to the entire number line, rather than remaining as two visibly separate pieces.

All-Real Disjunctive Result describes the special case of Overlapping Disjunctive Regions in which the two individual solution rays, taken together, cover every real number with no gap between them, so that the disjunctive inequality is satisfied by literally every possible value, mirroring the All-Real Inequality Result described for a single inequality reducing to a true numerical statement, but arising here from the union of two rays rather than from variable cancellation.

Disjunctive Interval Union is the concluding notation expressing the combined solution set, written using the same interval conventions established in One-Sided Interval Form for each individual ray, joined together with a symbol indicating their union whenever Disjoint Ray Result applies, or simplified to a single interval, or to the notation for all real numbers, whenever Overlapping Disjunctive Regions or All-Real Disjunctive Result renders the two pieces continuous or all-encompassing.