1.24 Inequality Representation Definitions

Inequality Representation Definitions establishes the vocabulary for distinguishing the different types of inequality relations by whether they include their boundary value, and describes how that boundary value is represented visually and conceptually depending on whether it is included in or excluded from the solution set.

Strict Inequality

A strict inequality is an inequality relation that excludes the exact boundary value from its solution set, using only the symbols < (less than) or > (greater than). The strict inequality x > 4 includes every real number greater than 4, but does not include 4 itself, since 4 is not strictly greater than 4.

Inclusive Inequality

An inclusive inequality is an inequality relation that includes the exact boundary value within its solution set, using the symbols ≤ (less than or equal to) or ≥ (greater than or equal to). The inclusive inequality x ≥ 4 includes every real number greater than 4 as well as 4 itself, since 4 satisfies "greater than or equal to 4."

Inequality Boundary

The inequality boundary is the specific numerical value that separates the values satisfying an inequality from the values that do not, marking the point on the number line where the inequality relation transitions from true to false or from false to true. In both x > 4 and x ≥ 4, the inequality boundary is the number 4, even though the two inequalities differ in whether that boundary value itself belongs to the solution set.

Open Boundary

An open boundary is the graphical and conceptual representation of an inequality boundary that is excluded from the solution set, corresponding to a strict inequality, and is typically drawn on a number line as an unfilled (open) circle at the boundary value, signaling that the boundary point itself is approached but never included.

Closed Boundary

A closed boundary is the graphical and conceptual representation of an inequality boundary that is included in the solution set, corresponding to an inclusive inequality, and is typically drawn on a number line as a filled (closed) circle at the boundary value, signaling that the boundary point itself is part of the solution.

Together, these definitions distinguish inequality relations by whether they include their boundary value—strict inequalities excluding it, inclusive inequalities including it—and establish the corresponding open and closed boundary representations used to display that distinction visually on a number line, ensuring the graphical picture of a solution set matches its symbolic meaning exactly.