Non-Strict Inequality
Pre-AlgebraA non-strict inequality uses <= or >= 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). The boundary value IS included in the solution.
Example
$x \ge 5$ means $x$ can be $5$, $6$, $7$, or any number $5$ or more. On a number line, you draw a filled (solid) circle at $5$.
Key Insight
The "or equal to" part is the key difference from a strict inequality. It lets the boundary value be part of the solution.
Definition
A non-strict inequality ($\le$ or $\ge$) includes the boundary point in its solution set. In interval notation, the included boundary is shown with a square bracket. On a number line, it is shown with a closed (filled) circle.
Example
$x \le 7$ has solution $(-\infty, 7]$, which includes $7$. A real-world example: a speed limit of $65$ mph means $x \le 65$; driving at exactly $65$ is permitted.
Key Insight
Non-strict inequalities produce closed sets on the number line. Many real-world constraints (minimum age, maximum load, qualifying scores) use non-strict inequalities because meeting the exact boundary is acceptable.
Definition
In an ordered field, $a \le b$ means $a < b$ or $a = b$, equivalently $b - a \ge 0$ (non-negative). Non-strict inequalities define closed sets in the standard topology on $\mathbb{R}$: the closed interval $[a, b] = \{x : a \le x \le b\}$ is the intersection of two closed half-lines. In optimization, constraints of the form $g(x) \le 0$ (non-strict) define closed feasible regions.
Example
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. By the Extreme Value Theorem, any continuous function on it achieves a maximum and minimum.
Key Insight
Compactness (which requires closedness in $\mathbb{R}^n$) is the property that guarantees maxima and minima exist. Non-strict inequalities produce the closed sets needed for this guarantee, which is why optimization problems typically use $\le$ constraints.