28.7 Linear Expressions within Absolute Value

Linear Expressions within Absolute Value extends every dual-branch and compound-inequality technique already developed to the general case in which the expression enclosed by the absolute value bars is a full linear expression, involving both a coefficient and a constant, rather than the bare variable used to introduce these techniques in their simplest form.

Shifted Absolute Value Expression describes an enclosed expression in which a constant has been added to or subtracted from the variable, such as the variable plus or minus a fixed amount, shifting the point at which the expression equals zero away from the origin itself and correspondingly shifting the center of any resulting solution set along the number line.

Scaled Linear Interior Expression describes an enclosed expression in which the variable is additionally multiplied by a coefficient other than one, stretching or compressing how quickly the enclosed expression's value changes as the variable changes, and correspondingly affecting the width of any resulting solution interval or the spacing of any resulting exterior rays relative to what a bare variable would produce.

Absolute Value Center Identification is the action of determining the specific value of the variable that makes the enclosed linear expression equal to exactly zero, found by solving the enclosed expression set equal to zero on its own, and identifying this value as the center around which the geometric interpretation of the absolute value relation, whether an equation or an inequality, is organized.

Positive Radius Identification is the action of confirming the numerical value on the opposite side of the isolated absolute value expression, verifying that it is indeed positive before proceeding with the standard dual-branch or compound-inequality techniques, matching Absolute Value Equal to a Positive Number for equations or the corresponding radius values used in Interior Absolute Value Inequalities and Exterior Absolute Value Inequalities.

Linear Branch Equation Resolution is the application of Enclosed Expression Equal to the Positive Radius and Enclosed Expression Equal to the Negative Radius to a Shifted Absolute Value Expression or Scaled Linear Interior Expression, requiring that each resulting branch equation be solved using Multi-Step Solving Sequence rather than a single inverse operation, since the presence of both a coefficient and a constant means each branch typically requires more than one step to isolate the variable.

Interior Compound Resolution is the application of Chained Inequality Resolution to the compound inequality produced from a Distance Less Than a Positive Radius or Distance at Most a Positive Radius relation involving a Shifted Absolute Value Expression or Scaled Linear Interior Expression, requiring that Same Additive Change across Three Parts and either Same Positive Scale across Three Parts or Negative Scaling across Three Parts be applied across all three parts of the chain to fully isolate the variable in the middle position.

Exterior Branch Resolution is the application of Independent Disjunctive Resolution to the pair of inequalities produced from a Distance Greater Than a Positive Radius or Distance at Least a Positive Radius relation involving a Shifted Absolute Value Expression or Scaled Linear Interior Expression, requiring each of the two branches, Left Exterior Inequality Branch and Right Exterior Inequality Branch, to be solved independently using whichever standard multi-step technique the specific enclosed expression demands, including careful attention to Relation Reversal after Negative Scaling if either branch requires division or multiplication by a negative coefficient.