Compound Inequality
A compound inequality joins two inequalities with "and" or "or," representing either an intersection or a union of solution sets.
Definition
A compound inequality combines two inequalities into one statement using the word "and" or "or": "and" means both must be true at the same time, and "or" means at least one must be true. "And" compound inequalities are written as a double inequality, $a < x < b$, and always form a bounded interval (unless there is no solution), while "or" compound inequalities are written as two separate statements and always form an unbounded set (two rays or the entire number line). This corresponds to a set operation on solution sets: "and" is set intersection, "or" is set union; in interval notation, intersections of intervals can always be expressed as a single interval (or the empty set), while unions may require multiple interval pieces, and in measure theory such sets are called Borel sets.
Example
"And" example: $2 < x < 8$ means $x$ is greater than $2$ AND less than $8$. "Or" example: $x < 1$ or $x > 5$ means $x$ is either less than $1$ or greater than $5$, written in interval notation as $(-\infty, 1) \cup (5, \infty)$. Solving $-3 < 2x + 1 \le 7$: subtract $1$ to get $-4 < 2x \le 6$, then divide by $2$ to get $-2 < x \le 3$, an interval $(-2, 3]$; and $\{x : x > -1\} \cap \{x : x \le 4\} = (-1, 4]$, while $\{x : x < 2\} \cup \{x : x \ge 6\} = (-\infty, 2) \cup [6, \infty)$, a disconnected set.
Key Insight
"And" inequalities describe a range between two values, while "or" inequalities describe two separate directions spreading away from a gap. The algebra of open and closed intervals under union and intersection generates the Borel sigma-algebra on $\mathbb{R}$, the foundational structure for measure theory and probability, so compound inequalities are the elementary building blocks of this hierarchy.