Strict Inequality
Pre-AlgebraA 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 answer.
Example
$x > 4$ is strict: $x$ can be $5$, $6$, or $4.1$, but NOT $4$ itself. On a number line, you draw an open (hollow) circle at $4$.
Key Insight
"Strict" means no wiggle room at the boundary. You must be strictly bigger or strictly smaller.
Definition
A strict inequality uses $<$ or $>$ (without the equals bar) to express that two quantities cannot be equal. The solution set is an open interval on the number line, represented with parentheses in interval notation and an open circle on a graph.
Example
$x < -2$ has solution set $(-\infty, -2)$ in interval notation. The boundary point $-2$ is excluded (open circle). Compare to $x \le -2$, which has solution $(-\infty, -2]$ and includes $-2$.
Key Insight
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 it does.
Definition
In an ordered field, $a < b$ (strict) 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.
Example
The open interval $(a, b) = \{x \in \mathbb{R} : a < x < b\}$ is defined by two strict inequalities. Every open set in the standard topology on $\mathbb{R}$ is a countable union of open intervals.
Key Insight
The topology generated by strict inequalities (open intervals) is the foundation of real analysis. The distinction between open and closed sets ultimately determines continuity, compactness, and completeness.