Inequality Symbols
Inequality symbols are the mathematical notation used to compare two quantities: less than ($<$), greater than ($>$), less than or equal to ($\le$), greater than or equal to ($\ge$), and not equal to ($\neq$).
Definition
Inequality symbols show how two numbers or expressions compare. The five symbols are $<$ (strictly less than), $>$ (strictly greater than), $\le$ (less than or equal to), $\ge$ (greater than or equal to), and $\neq$ (not equal). On a number line, $<$ and $>$ use open circles since the boundary is not included, while $\le$ and $\ge$ use closed circles since the boundary is included. More formally, inequality symbols express the total order relation on ordered fields such as $\mathbb{R}$: a total order satisfies trichotomy (exactly one of $a < b$, $a = b$, $a > b$ holds), transitivity ($a < b$ and $b < c$ imply $a < c$), and compatibility with addition and multiplication; not all fields can be ordered, for example $\mathbb{C}$ cannot be ordered consistently with its field structure, since $i^2 = -1$ would violate closure of the positive cone.
Example
$3 < 7$ means $3$ is less than $7$, $10 > 4$ means $10$ is greater than $4$, and $x \ge 5$ means $x$ is at least $5$. Graphing $x > 3$ uses an open circle at $3$ with an arrow pointing right; graphing $x \le -1$ uses a closed circle at $-1$ with an arrow pointing left. In the ordered field $\mathbb{R}$, the strict order $<$ is defined by $a < b$ iff $b - a$ is a positive real number, a definition that generalizes to ordered fields and ordered rings.
Key Insight
A memory trick: the inequality symbol is like an arrow or an alligator mouth that always opens toward the bigger number. The distinction between strict ($<$, $>$) and non-strict ($\le$, $\ge$) inequalities determines whether the boundary point is part of the solution, a detail that matters in optimization and real analysis, and the fact that not every field can be ordered (such as $\mathbb{C}$) shows just how special the ordered structure of $\mathbb{R}$ really is.