Quadratic Inequalities

A quadratic inequality is a mathematical statement that compares a quadratic expression of the form $ax^2+bx+c$ to another expression or value using the symbols $<$, $>$, $\leq$, or $\geq$. A quadratic inequality can be expressed in one of these forms, where $a\neq0$:

• $ax^2+bx+c<0$
• $ax^2+bx+c>0$
• $ax^2+bx+c\leq 0$
• $ax^2+bx+c\geq 0$

A solution of an inequality is any value of the variable that makes the inequality true, which means the comparison is valid. A quadratic inequality may have zero, one, or infinitely many real solutions. When there are infinitely many solutions, they may be represented on a number line or by using set builder notation (a shorthand way of writing a set of values).

The number of solutions of a quadratic inequality can be determined by looking at the graph. For example, the solutions of $ax^2+bx+c<0$ are the $x$-values where the graph of $f(x)=ax^2+bx+c$ is below the $x$-axis. If the graph has 2 $x$-intercepts and the curve opens up, the solutions are all of the values between the $x$-intercepts, so there are infinitely many solutions.

To solve a quadratic inequality, the solution set may be found by graphing the related function and determining which intervals of the domain satisfy the inequality.