Inequality
Pre-AlgebraAn 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 that compares two values using symbols that mean "less than," "greater than," "less than or equal to," or "greater than or equal to." Instead of saying two things are equal, it says one is bigger or smaller.
Example
$x > 5$ means $x$ is greater than $5$. Any number bigger than $5$ (like $6$, $7$, or $100$) makes this true. The solution is not just one number but a whole range.
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."
Definition
An inequality is a relation between two expressions using one of the symbols $<$, $>$, $\le$, or $\ge$. The solution set is typically a range of values rather than a single number. When multiplying or dividing both sides by a negative number, the inequality symbol must be flipped.
Example
Solve $3x - 4 > 8$. Add $4$: $3x > 12$. Divide by $3$: $x > 4$. Solution set: all real numbers greater than $4$, written $(4, \infty)$ in interval notation.
Key Insight
The rule about flipping the inequality when multiplying or dividing by a negative is the most commonly forgotten rule. Example: $-2x < 6$ gives $x > -3$, not $x < -3$.
Definition
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. Systems of linear inequalities define convex feasible regions, the foundation of linear programming.
Example
The inequality $|x - 3| < 2$ is equivalent to $-2 < x - 3 < 2$, which gives $1 < x < 5$ and defines an open interval $(1, 5)$. The distance interpretation links absolute value to metric spaces.
Key Insight
Linear programming (LP) maximizes or minimizes a linear objective over a convex polyhedron defined by linear inequalities. LP is the backbone of operations research, logistics optimization, and machine learning regularization.