Compound Inequality
Pre-AlgebraA 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. "Or" means at least one must be true.
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 $5$.
Key Insight
"And" inequalities describe a range between two values. "Or" inequalities describe two separate directions spreading away from a gap.
Definition
A compound inequality uses "and" (intersection) or "or" (union) to join two simple inequalities. "And" compound inequalities are written as a double inequality: $a < x < b$. "Or" compound inequalities are written as two separate statements.
Example
"And": solve $-3 < 2x + 1 \le 7$. Subtract $1$: $-4 < 2x \le 6$. Divide by $2$: $-2 < x \le 3$. Interval: $(-2, 3]$. "Or": $x < -1$ or $x > 4$ written in interval notation is $(-\infty, -1) \cup (4, \infty)$.
Key Insight
"And" compounds always form a bounded interval (unless they produce no solution). "Or" compounds always form an unbounded set (two rays or the entire number line).
Definition
A compound inequality 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. In measure theory, such sets are Borel sets.
Example
$\{x : x > -1\} \cap \{x : x \le 4\} = \{x : -1 < x \le 4\} = (-1, 4]$. $\{x : x < 2\} \cup \{x : x \ge 6\} = (-\infty, 2) \cup [6, \infty)$, a disconnected set.
Key Insight
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. Compound inequalities are the elementary building blocks of this hierarchy.