1.60 Quadratic Inequality Definitions
Quadratic inequalities compare quadratic expressions to values using greater than or less than.
Quadratic Inequality Definitions establishes the vocabulary for inequalities built from a quadratic expression rather than a linear one, describing the specific values that separate the number line into distinct regions of behavior for such an inequality, and the intervals produced by those separating values within which the quadratic expression maintains a single consistent sign.
Quadratic Inequality
A quadratic inequality is an inequality involving a quadratic expression compared to zero or to another expression, using <, >, ≤, or ≥, such as x² − x − 6 > 0. Unlike a linear inequality, whose solution set is always a single unbounded or bounded interval, a quadratic inequality's solution set can consist of one bounded interval, one or two unbounded intervals, or a combination of intervals, depending on how the underlying parabola behaves relative to the horizontal axis.
Quadratic Boundary Value
A quadratic boundary value is a solution to the related quadratic equation formed by replacing a quadratic inequality's inequality symbol with an equals sign, marking a point where the quadratic expression's value is exactly zero and where its sign may change from positive to negative or from negative to positive. The quadratic inequality x² − x − 6 > 0 has the related equation x² − x − 6 = 0, whose solutions, x = 3 and x = −2, are the quadratic boundary values that divide the number line into regions to be tested.
Quadratic Sign Interval
A quadratic sign interval is one of the regions of the number line created when the quadratic boundary values divide it into separate sections, within which the quadratic expression's value maintains a single consistent sign—entirely positive or entirely negative—throughout that entire section. Because a quadratic boundary value is the only place where the expression's sign can change, testing a single representative point within each quadratic sign interval reveals the sign of the entire interval, and comparing that sign against the original inequality determines whether the interval belongs to the solution set.
Together, these definitions describe the standard method for solving a quadratic inequality: locating its quadratic boundary values from the related equation, using those values to divide the number line into quadratic sign intervals, and testing each interval to determine which ones satisfy the original inequality and therefore belong to the final solution set.