62 Quadratic Inequalities

Quadratic Inequalities is the study of solving inequalities involving a quadratic expression, using the expression's boundary values (found by solving the corresponding quadratic equation) to divide the number line into intervals, then testing each interval to determine where the inequality holds true.

The Scope of Quadratic Inequalities

A quadratic inequality compares a quadratic expression to zero (or, after rearrangement, to another expression) using <, >, ≤, or ≥, such as x² - x - 6 > 0. Unlike a linear inequality, whose solution set is always a single interval or ray, a quadratic inequality's solution set can consist of a bounded interval, two separate unbounded rays, or the entire real number line (or none of it at all), depending on the shape of the associated parabola and the direction of the inequality.

Preparing a Quadratic Inequality

Solving a quadratic inequality begins by rearranging it so that one side is zero, exactly as preparing a quadratic equation for factoring, moving every term to one side and simplifying.

Finding Quadratic Boundary Values

The boundary values of a quadratic inequality are found by solving the corresponding quadratic equation, obtained by temporarily replacing the inequality symbol with an equals sign, using factoring, the square root method, completing the square, or the quadratic formula as appropriate.

These boundary values are the points where the quadratic expression equals exactly zero, and they divide the number line into distinct intervals within which the expression's sign cannot change.

Analyzing the Sign of Each Interval

Once the boundary values are found, they split the number line into intervals — in the example above, three intervals: values less than -2, values between -2 and 3, and values greater than 3. A test point from within each interval is substituted into the original quadratic expression to determine whether that entire interval produces a positive or negative result, since a continuous quadratic expression's sign can only change at its boundary values.

Testing x = -3 (from the leftmost interval) gives a positive result, testing x = 0 (from the middle interval) gives a negative result, and testing x = 4 (from the rightmost interval) gives a positive result, showing the expression alternates sign at each boundary value, as expected for a quadratic with two distinct real roots.

Assembling the Solution Set

The final solution set of a quadratic inequality consists of every interval whose test point satisfied the original inequality. For x² - x - 6 > 0, the intervals producing a positive result — values less than -2 and values greater than 3 — form the solution, written as x < -2 or x > 3, since the inequality is strict and does not include the boundary values themselves. If the original inequality had used ≥ instead of >, the boundary values would be included, using closed circles on the number line and square brackets in interval notation.

Interpreting the Solution Through the Parabola's Graph

Because a quadratic expression's sign corresponds directly to whether its graph lies above or below the x-axis, a quadratic inequality can also be solved, or its solution verified, by considering the shape of the associated parabola: solving "expression > 0" corresponds to finding the x-values where the parabola lies above the x-axis, and solving "expression < 0" corresponds to finding where it lies below. This graphical view explains directly why an upward-opening parabola with two real roots produces a solution set of two unbounded rays for a ">" inequality and a single bounded interval for a "<" inequality, and why the opposite pattern occurs for a downward-opening parabola.

Diagnosing Errors in Quadratic Inequalities

Common errors in this area include selecting a test point that does not lie strictly within its intended interval, particularly one accidentally chosen equal to a boundary value, forgetting to include or exclude the boundary values correctly based on whether the original inequality was strict or inclusive, testing only one interval and assuming its result applies to every interval rather than checking each one independently, and misinterpreting the graphical relationship between a parabola's position relative to the x-axis and the direction of the original inequality symbol.