28.6 Exterior Absolute Value Inequalities

Exterior Absolute Value Inequalities are absolute value inequalities in which the isolated absolute value expression is compared using a greater-than or greater-than-or-equal symbol against a positive value, producing a solution set consisting of every value whose distance from zero exceeds a stated amount, translating directly into a disjunctive compound inequality of the kind addressed in Disjunctive Inequalities.

Distance Greater Than a Positive Radius describes the strict version of this inequality, shown in the general form below.

where a is positive, expressing, following Nonnegative Distance Interpretation, that the enclosed expression must lie strictly farther than the given distance from zero, excluding both the case of lying exactly that distance away and every case of lying closer.

Distance at Least a Positive Radius describes the inclusive version of this inequality, replacing the strict comparison with a greater-than-or-equal symbol, expressing that the enclosed expression must lie that given distance from zero or farther, permitting the boundary distance itself to be included among the satisfying values.

Left Exterior Inequality Branch is the first of the two branches derived from this structure, expressing that the enclosed linear expression is less than the negative of the stated radius, corresponding to values lying far to the left of zero, beyond the negative boundary.

Right Exterior Inequality Branch is the second of the two branches, expressing that the enclosed linear expression is greater than the positive radius itself, corresponding to values lying far to the right of zero, beyond the positive boundary.

Strict Exterior Ray Pair is the pair of inequalities produced when Distance Greater Than a Positive Radius is translated, joining Left Exterior Inequality Branch and Right Exterior Inequality Branch with the word "or" using strict comparisons throughout, shown in the general form below.

This translation directly reflects the geometric fact that every value lying strictly farther than a given distance from zero must lie either strictly beyond the negative boundary or strictly beyond the positive boundary.

Inclusive Exterior Ray Pair is the corresponding pair of inequalities produced when Distance at Least a Positive Radius is translated, replacing both strict comparisons with inclusive ones so that each boundary value itself is included in its respective branch.

Disjunctive Absolute Inequality Result is the concluding classification confirming that every Exterior Absolute Value Inequalities case, once translated into its equivalent pair of branches, is solved and represented using precisely the same techniques already established for Disjunctive Inequalities generally, including Independent Disjunctive Resolution, Combined Outer Solution Regions, and the resulting Disjoint Ray Result, together with the appropriate graphical or interval notation reflecting whether Strict Exterior Ray Pair or Inclusive Exterior Ray Pair governs the inclusion of each boundary.