Strict Inequality
A strict inequality uses < or > to show that one quantity is strictly less than or strictly greater than another, not allowing equality.
Definition
A strict inequality uses $<$ or $>$ and means "but NOT equal to": the boundary value is not included in the solution, so on a number line you draw an open (hollow) circle at that value. The solution set of a strict inequality is an open interval, represented with parentheses in interval notation. Formally, in an ordered field, $a < b$ means $b - a$ belongs to the positive cone $P$, where $P$ is closed under addition and multiplication, contains no zero, and satisfies the trichotomy law; strict inequalities define open sets in the order topology on $\mathbb{R}$, forming a basis for the standard topology, and this topology generated by open intervals is the foundation of real analysis, since the distinction between open and closed sets determines continuity, compactness, and completeness.
Example
$x > 4$ is strict: $x$ can be $5$, $6$, or $4.1$, but NOT $4$ itself, drawn with an open circle at $4$. $x < -2$ has solution set $(-\infty, -2)$, excluding the boundary point $-2$, compared to $x \le -2$, which includes $-2$ and has solution $(-\infty, -2]$. The open interval $(a, b) = \{x \in \mathbb{R} : a < x < b\}$ is defined by two strict inequalities, and every open set in the standard topology on $\mathbb{R}$ is a countable union of such open intervals.
Key Insight
"Strict" means no wiggle room at the boundary: you must be strictly bigger or strictly smaller. Open versus closed boundaries are critical in calculus and real analysis; for example, a function defined on $(0, 1)$ (open) may not achieve its maximum, whereas on $[0, 1]$ (closed) the Extreme Value Theorem guarantees that it does.