1.28 Absolute Value Relation Definitions

Absolute Value Relation Definitions establishes the vocabulary for equations and inequalities that involve an absolute value expression, describing the two-part structure such relations imply, the center point and radius that define the symmetric range of values they describe, and the case-splitting technique required to resolve them into ordinary equations or inequalities without absolute value bars.

Absolute Value Equation

An absolute value equation is an equation in which the variable expression appears inside absolute value bars set equal to some value, such as |x − 3| = 5. Because absolute value strips away sign, an absolute value equation set equal to a positive value generally corresponds to two distinct possibilities for the expression inside the bars—one positive and one negative—each of which must be solved separately to find every solution.

Absolute Value Inequality

An absolute value inequality is an inequality in which the variable expression appears inside absolute value bars compared to some value using <, >, ≤, or ≥, such as |x − 3| < 5 or |x − 3| > 5. Depending on whether the inequality symbol indicates "less than" or "greater than," the absolute value inequality describes either a single bounded range of values or two separate unbounded ranges.

Absolute Value Center

The absolute value center is the value being subtracted from the variable inside the absolute value bars, representing the point on the number line around which the relation's solution values are symmetrically arranged. In the expression |x − 3|, the absolute value center is 3, since the expression measures how far x lies from that specific point.

Absolute Value Radius

The absolute value radius is the value on the opposite side of the equation or inequality from the absolute value expression, representing the maximum (or minimum) allowed distance from the absolute value center. In |x − 3| < 5, the absolute value radius is 5, meaning the relation describes every value of x lying within a distance of 5 from the center point 3.

Absolute Value Case Split

The absolute value case split is the technique of rewriting an absolute value equation or inequality as two separate ordinary equations or inequalities with the bars removed—one covering the case where the inner expression is nonnegative, and one covering the case where the inner expression is negative—since the two cases require opposite treatment of the expression's sign. Applying a case split to |x − 3| = 5 produces the two equations x − 3 = 5 and x − 3 = −5, which solve to x = 8 and x = −2 respectively.

Together, these definitions describe absolute value relations as symmetric conditions built around a center point and a radius, and establish the case split as the essential technique for converting the absolute-value-bar form into ordinary equations or inequalities that can be solved using the isolation methods already established for linear equations and inequalities.