Inequality Symbols
Pre-AlgebraInequality symbols are the mathematical notation used to compare two quantities: less than (<), greater than (>), less than or equal to (<=), greater than or equal to (>=), and not equal to (not equal).
Definition
Inequality symbols show how two numbers or expressions compare. The main symbols are: $<$ (less than), $>$ (greater than), $\le$ (less than or equal to), and $\ge$ (greater than or equal to).
Example
$3 < 7$ means $3$ is less than $7$. $10 > 4$ means $10$ is greater than $4$. $x \ge 5$ means $x$ is at least $5$.
Key Insight
A memory trick: the inequality symbol is like an arrow or an alligator mouth that always opens toward the bigger number.
Definition
The five inequality 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 (boundary not included), while $\le$ and $\ge$ use closed circles (boundary included).
Example
Graphing $x > 3$: open circle at $3$, arrow pointing right. Graphing $x \le -1$: closed circle at $-1$, arrow pointing left.
Key Insight
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.
Definition
Inequality symbols formalize the total order relation on ordered fields such as $\mathbb{R}$. A total order satisfies: (1) trichotomy: exactly one of $a < b$, $a = b$, $a > b$ holds; (2) transitivity: $a < b$ and $b < c$ imply $a < c$; (3) compatibility with addition and multiplication. The not-equal symbol $\neq$ denotes the complement of the equality relation.
Example
In the ordered field $\mathbb{R}$, the strict order $<$ is defined by: $a < b$ iff $b - a$ is a positive real number (a member of the positive cone). This definition generalizes to ordered fields and ordered rings.
Key Insight
Not all fields can be ordered. For example, $\mathbb{C}$ cannot be ordered consistently with its field structure. An ordered field must have $-1$ as a sum of squares of negatives, ruling out $\mathbb{C}$ where $i^2 = -1$ is negative, violating closure of the positive cone.