Quadratic inequalities and systems of inequalities

A quadratic inequality is an inequality that involves a quadratic expression, such as $ax^2 + bx + c > 0$, $ax^2 + bx + c < 0$, $ax^2 + bx + c \geq 0$, or $ax^2 + bx + c \leq 0$. To solve a quadratic inequality, we find the values of $x$ that make the inequality true. This usually involves finding the roots of the related quadratic equation and analyzing the sign of the expression in different intervals.

Important properties

• The solution set of a quadratic inequality is usually an interval or a union of intervals on the real number line.

• The sign of the quadratic expression changes at its roots (where $ax^2 + bx + c = 0$).

• If $a > 0$, the parabola opens upwards; if $a < 0$, it opens downwards. This affects which intervals satisfy the inequality.

• Systems of inequalities involve finding values that satisfy more than one inequality at the same time, often by finding the intersection of solution sets.