Non-Strict Inequality
A non-strict inequality uses $\le$ or $\ge$ to show that one quantity is less than or equal to, or greater than or equal to another, including the possibility of equality.
Definition
A non-strict inequality uses $\le$ (less than or equal to) or $\ge$ (greater than or equal to), and the boundary value IS included in the solution: on a number line you draw a filled (solid) circle at that value, and in interval notation the included boundary is shown with a square bracket. Formally, in an ordered field, $a \le b$ means $a < b$ or $a = b$, equivalently $b - a \ge 0$; non-strict inequalities define closed sets in the standard topology on $\mathbb{R}$, and in optimization, constraints of the form $g(x) \le 0$ define closed feasible regions. Compactness, which requires closedness in $\mathbb{R}^n$, is the property that guarantees maxima and minima exist, which is why optimization problems typically use $\le$ constraints.
Example
$x \ge 5$ means $x$ can be $5$, $6$, $7$, or any number $5$ or more, drawn with a filled circle at $5$. $x \le 7$ has solution $(-\infty, 7]$, which includes $7$; a real-world example is a speed limit of $65$ mph, meaning $x \le 65$, where driving at exactly $65$ is permitted. The closed unit disk $\{(x, y) : x^2 + y^2 \le 1\}$ is a compact, closed subset of $\mathbb{R}^2$ defined by a non-strict inequality, and by the Extreme Value Theorem, any continuous function on it achieves a maximum and minimum.
Key Insight
The "or equal to" part is the key difference from a strict inequality, letting the boundary value be part of the solution; many real-world constraints (minimum age, maximum load, qualifying scores) use non-strict inequalities because meeting the exact boundary is acceptable, and it is exactly this closedness that guarantees optimization problems have solutions.