Quadratic inequalities

A quadratic inequality is solved by looking at where the graph of a quadratic function lies above or below the $x$-axis. Let us solve $x^2 - x - 6 > 0$.

First, at equality, $x^2 - x - 6 = 0$ gives $(x - 3)(x + 2) = 0$, so $x = 3, -2$. These are where the parabola $y = x^2 - x - 6$ meets the $x$-axis, and they split the number line into three ranges.

$x^2 - x - 6 > 0$ means "$y$ is positive," that is the graph is above the $x$-axis. Since this parabola opens upward, it is above the axis outside the two intersection points and below between them. So the solution is $x < -2$ or $x > 3$.

Conversely, $x^2 - x - 6 < 0$ holds where the graph is below the $x$-axis, the inside range $-2 < x < 3$.

For an upward-opening parabola you can remember: "$> 0$ means outside the two roots, $< 0$ means between them." The large dots on the graph are the boundaries $(3, 0)$ and $(-2, 0)$, and outside them the parabola lies above the $x$-axis.