25.4 Multi-Step Linear Inequalities

Multi-Step Linear Inequalities are one-variable linear inequalities that, like their equation counterparts, require simplification and multiple inverse operations before the variable is isolated, combining Additive Inequality Transformations and Multiplicative Inequality Transformations in sequence while remaining alert throughout to Relation Reversal after Negative Scaling wherever it applies.

Independent Inequality-Side Reduction is the opening action of the solving sequence, reducing the left side and the right side of the inequality separately to their simplest forms before any term is moved across the inequality symbol, mirroring Independent Side Simplification from two-sided equation solving but applied to the two sides of a comparison rather than an equality.

Group Expansion within an Inequality is the action of applying the distributive property to any parenthetical grouping present on either side of the inequality, following the identical technique used in Positive Outer Factor Distribution and Negative Outer Factor Distribution, expanding an outer factor across every term inside the grouping before further reduction proceeds.

Within-Side Like-Term Reduction is the action of combining like terms, whether variable terms or constants, on each side individually once any necessary expansion has been completed, reducing each side to at most one combined variable term and one combined constant term, exactly as in Equation-Side Simplification.

Opposite-Side Variable Collection is the action, required when the variable appears on both sides of the inequality, of moving the variable term with the smaller coefficient to the side of the other variable term using Additive Inequality Transformations, consolidating the variable onto a single side in a manner directly analogous to Variable-Term Consolidation Step, and preserving the inequality's direction throughout since only addition or subtraction is involved at this stage.

Opposite-Side Constant Collection is the corresponding action of moving the remaining constant term attached to the variable's side to the opposite side, again using Additive Inequality Transformations, directly analogous to Constant-Term Consolidation Step, and likewise preserving the inequality's direction since this step, too, involves only addition or subtraction.

Final Inequality Coefficient Step is the concluding transformation, applying Multiplicative Inequality Transformations to remove the coefficient remaining on the isolated variable term, dividing or multiplying both sides by that coefficient. This step demands the greatest care in the entire sequence, since it is here that Relation Reversal after Negative Scaling must be correctly applied if the coefficient is negative, or correctly withheld if the coefficient is positive, following Positive Scaling without Relation Reversal.

One-Sided Solution Inequality is the resulting statement once Final Inequality Coefficient Step is complete: the variable stands alone with a coefficient of exactly one, compared through an inequality symbol, whose direction reflects any reversal applied during the final step, to a single boundary value. This resulting inequality matches the Single-Boundary Solution Form expected for this scope and can be expressed through any of the representations established in Inequality Solution Set Notation or graphed directly using Number-Line Solution Representation.