28.5 Interior Absolute Value Inequalities

Interior Absolute Value Inequalities are absolute value inequalities in which the isolated absolute value expression is compared using a less-than or less-than-or-equal symbol against a positive value, producing a solution set consisting of every value whose distance from zero falls within a bounded range, translating directly into a conjunctive compound inequality of the kind addressed in Conjunctive Inequalities.

Distance Less 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 less than the given distance from zero, excluding both the case of lying exactly that distance away and every case of lying farther away.

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

Strict Interior Compound Inequality is the equivalent chained inequality produced by translating Distance Less Than a Positive Radius, expressing that the enclosed expression is simultaneously greater than the negative of the radius and less than the positive radius itself, shown in the general form below.

This translation directly reflects the geometric fact that every value lying strictly less than a given distance from zero must lie strictly between the negative and positive versions of that distance.

Inclusive Interior Compound Inequality is the corresponding equivalent chained inequality produced by translating Distance at Most a Positive Radius, replacing both strict comparisons in the chain with inclusive ones, expressing that the enclosed expression lies between the negative and positive radius inclusive of both boundary values themselves.

Centered Bounded Interval is the resulting solution set once the compound inequality has been solved using Chained Inequality Resolution, forming a Bounded Interval Result whose midpoint corresponds to the point where the enclosed linear expression would equal zero, extending an equal distance in both directions up to the radius value, mirroring the symmetric structure already established through Symmetric Point Distances.

Interior Endpoint Inclusion is the determination of whether the two boundary values of the Centered Bounded Interval are themselves included in the solution set, governed directly by whether the original inequality used a strict or inclusive comparison, following Strict Interior Compound Inequality for an excluded boundary or Inclusive Interior Compound Inequality for an included boundary.

Conjunctive Absolute Inequality Result is the concluding classification confirming that every Interior Absolute Value Inequalities case, once translated into its equivalent chained form, is solved and represented using precisely the same techniques already established for Conjunctive Inequalities generally, including Chained Inequality Resolution, Bounded Interval Result, and the appropriate graphical or interval notation reflecting Interior Endpoint Inclusion at each of the two boundaries.