26.3 Chained Inequality Resolution

Chained Inequality Resolution is the technique of solving a compound inequality written in Chained Conjunctive Form by applying each transformation simultaneously to all three parts of the chain, the left boundary expression, the middle variable expression, and the right boundary expression, rather than separating the chain into two independent inequalities to be solved one at a time.

Same Additive Change across Three Parts is the action of applying an Inequality Addition Transformation or Inequality Subtraction Transformation identically to the left part, the middle part, and the right part of the chain all at once, mirroring the requirement that additive transformations be applied to both sides of an ordinary inequality, but extended here to all three components of the chain simultaneously in order to preserve the validity of the entire chained statement.

Same Positive Scale across Three Parts is the corresponding action of applying a Multiplication by a Positive Quantity or Division by a Positive Quantity identically across all three parts of the chain, following Positive Scaling without Relation Reversal, so that neither comparison symbol within the chain needs to be reversed.

Negative Scaling across Three Parts is the corresponding action taken when the chain must be multiplied or divided by a negative quantity, applying that negative scale factor identically to the left part, the middle part, and the right part, in accordance with Relation Reversal after Negative Scaling.

Reversal of Both Comparison Symbols is the essential consequence of Negative Scaling across Three Parts: because a chained inequality contains two comparison symbols, one connecting the left part to the middle part and one connecting the middle part to the right part, a negative scaling requires both of these symbols to be reversed simultaneously, not merely one of them, since the entire chain has been scaled by the identical negative factor.

Middle Variable Isolation is the goal toward which Same Additive Change across Three Parts and either Same Positive Scale across Three Parts or Negative Scaling across Three Parts are directed: through a sequence of transformations applied identically across all three parts, the middle part of the chain is reduced to the variable alone, with no other operation remaining attached to it.

Ordered Final Chain is the resulting statement once Middle Variable Isolation has been achieved: a chain of the same general form as the original, with the isolated variable in the middle position and two boundary values, one on the left and one on the right, connected by comparison symbols whose direction reflects any reversal required by Negative Scaling across Three Parts, shown in the general form below.

Chained Solution Interval is the interpretation of the Ordered Final Chain as a Bounded Interval Result, representing every value lying between the left boundary value and the right boundary value, with the inclusion or exclusion of each boundary determined by Open and Closed Compound Endpoints according to whether the original chain used strict or inclusive comparisons at each position, and expressible through One-Sided Interval Form extended to a two-boundary interval or through a Number-Line Solution Representation showing a bounded segment rather than an unbounded ray.