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.

The function is positive, or greater than zero, when $x<0.5$ or when $x>3$.

The solutions of $x^2+x+2<0$ are the values of $x$, where $x^2+x+2$ is less than zero. That means that the graph of $f(x)=x^2+x+2$ lies below the $x$-axis.

Since the graph is entirely above the $x$-axis, the inequality has no solutions.