A mathematical expression with an inequality symbol is called an inequality. The inequality symbols are: \[\begin{array}{|c|c|c|c|} \hline \lt & \text{less than} & \le & \text{less than or equal to} \\ \hline \gt & \text{greater than} & \ge & \text{greater than or equal to}\\ \hline\end{array}\]

A quadratic inequality with unknown \(\boldsymbol{x}\) is an inequality that can be reduced, by elementary operations, to a basic form \(ax^2+bx+c<0\), where \(a, b\) and \(c\) are numbers, and instead of \(\lt\) there may also be one of the other inequality symbols.

An elementary operation is understood to be expansion of brackets, regrouping of subexpressions, addition or subtraction of equivalent expressions on both sides of the equation, or multiplication or division on both sides of the equation by a number distinct from zero.

The solution of a quadratic inequality with unknown \(x\) is in the form \(x<C\), \(x\le C\), \(x\ge C\) or \(x\gt C\), where \(C\) is a number, and logical combinations hereof. These combinations are sometimes summarized again: \(x>1\;\wedge\; x<2\) can be written as \(1<x<2\).

Examples
\[\begin{aligned}x^2&<2x+3\\ &\phantom{wxyz} \blue{\text{quadratic inequality}} \\ x^2&-2x-3<0 \Leftrightarrow\\ &\hspace{-0.6cm}(x-1)^2-4<0\\ &\phantom{wxyz} \blue{\text{equivalent inequality}}\end{aligned} \]

\(\quad\) Inequality \(x^2<2x+3\) has
\(\quad\) solution \(-1<x< 3\).
\(\quad\) This follows from the below reduction: \[\begin{aligned} x^2&<2x+3 &\Leftrightarrow\\ x^2-2x-3&<0 &\Leftrightarrow\\ (x-1)^2-4&<0 &\Leftrightarrow\\ (x-1)^2&<4&\Leftrightarrow\\ -2<x-1&<2&\Leftrightarrow\\ -1<x&< 3\end{aligned}\]
\(\quad\) Inequality \(x^2+2x+1>9\) has
\(\quad\) solution \(x<-4\;\vee\;x>2\).
\(\quad\) This follows from the below reduction: \[\begin{aligned} x^2+2x+1&>9 &\Leftrightarrow\\ (x+1)^2&>3^2 &\Leftrightarrow\\ x+1<-3\;\vee\; x+1&>3 &\Leftrightarrow\\ x<-4\;\vee\; x&>2 \end{aligned}\]