Inequality
An inequality is a mathematical statement comparing two expressions using symbols such as less than, greater than, or not equal to.
Definition
An inequality is a math statement comparing two expressions using symbols meaning "less than," "greater than," "less than or equal to," or "greater than or equal to," written $<$, $>$, $\le$, or $\ge$; rather than saying two things are equal, it says one is bigger or smaller, and its solution set is typically a whole range of values rather than a single number. A key rule is that multiplying or dividing both sides by a negative number flips the inequality symbol. More formally, an inequality is a partial-order relation on expressions within an ordered field or ordered ring; for real-valued polynomial inequalities, Sturm's theorem provides an algorithm for counting real roots in an interval, and systems of linear inequalities define the convex feasible regions that are the foundation of linear programming.
Example
$x > 5$ means $x$ is greater than $5$: any number bigger than $5$, like $6$, $7$, or $100$, makes it true. Solving $3x - 4 > 8$: add $4$ to get $3x > 12$, then divide by $3$ to get $x > 4$, written $(4, \infty)$ in interval notation. The inequality $|x - 3| < 2$ is equivalent to $-2 < x - 3 < 2$, giving $1 < x < 5$, an open interval whose distance interpretation links absolute value to metric spaces.
Key Insight
Inequalities describe a range of possible answers, which is more realistic for many real-world situations, like "you must be at least $13$ years old." The rule about flipping the inequality when multiplying or dividing by a negative number is the most commonly forgotten rule (for example, $-2x < 6$ gives $x > -3$, not $x < -3$), and linear programming, which maximizes or minimizes a linear objective over a convex polyhedron defined by linear inequalities, is the backbone of operations research, logistics optimization, and machine learning regularization.