25.2 Additive Inequality Transformations

Additive Inequality Transformations are the operations of adding or subtracting the same quantity from both sides of an inequality, extending the addition and subtraction properties of equality to the setting of an inequality while preserving the truth and the direction of the comparison being expressed.

Inequality Addition Transformation is the action of adding an identical quantity to both sides of an inequality, producing a new inequality that retains the same comparison symbol as the original, shown in the general principle below.

This transformation holds true regardless of whether the quantity added is positive, negative, or zero, since adding the same amount to both sides shifts both quantities by an identical distance and cannot alter which one remains larger.

Inequality Subtraction Transformation is the corresponding action of subtracting an identical quantity from both sides of an inequality, likewise producing a new inequality that retains the same comparison symbol as the original. Like Inequality Addition Transformation, this operation holds regardless of the sign of the quantity subtracted, since subtracting the same amount from both sides preserves the relative distance and ordering between them.

Comparison Direction Preservation is the property, shared by both Inequality Addition Transformation and Inequality Subtraction Transformation, that the direction of the inequality symbol never changes as a result of these operations. This stands in direct contrast to the special caution required for multiplicative transformations by a negative quantity, and it means that additive operations can be applied to an inequality with the same straightforward confidence as they are applied to an equation.

Variable Isolation after Additive Change is the application of these transformations to a one-variable linear inequality in which the variable is connected to the rest of the inequality through addition or subtraction, mirroring Additive Layer Removal from equation solving: the constant attached to the variable is removed by adding or subtracting that same value from both sides, following whichever of Inequality Addition Transformation or Inequality Subtraction Transformation correctly reverses the operation originally attached to the variable.

Additive Inequality Solution Set is the resulting collection of values once Variable Isolation after Additive Change has been carried out, expressed as the isolated variable compared to a single boundary value with the same inequality symbol as the original, and representable through any of the forms established in Inequality Solution Set Notation, including Symbolic Inequality Form, One-Sided Interval Form, or a Number-Line Solution Representation, since Comparison Direction Preservation guarantees that the symbol obtained after isolation is the same symbol that governs the true solution set.