Understanding Interval Notation and Inequalities
An interval represents a continuous set of real numbers between two fixed values, known as endpoints. Intervals and inequalities describe identical mathematical relationships—both specify which numbers satisfy a given condition. For example, the inequality 2 < x < 5 defines exactly the same set as the interval (2, 5): all real numbers strictly between 2 and 5.
The relationship is direct and mechanical. Every inequality statement maps to a unique interval, and every interval corresponds to an inequality. Mastering this translation is essential for graphing, solving systems, and communicating mathematical constraints.
The Three Categories of Intervals
Intervals are classified by whether their endpoints are included in the set:
- Open intervals exclude both endpoints. Represented with parentheses: (a, b). Corresponds to a < x < b.
- Closed intervals include both endpoints. Represented with square brackets: [a, b]. Corresponds to a ≤ x ≤ b.
- Half-open intervals include exactly one endpoint. Written as (a, b] or [a, b). Correspond to a < x ≤ b or a ≤ x < b.
Unbounded intervals extend to infinity in one or both directions. For instance, [3, ∞) includes all numbers greater than or equal to 3, and (−∞, −2) includes all numbers less than −2.
Inequality to Interval Conversion Reference
Use this systematic table to convert any standard inequality into interval notation. The left column shows the inequality form, and the right shows its interval equivalent:
Inequality Interval
a < x < b (a, b)
a ≤ x ≤ b [a, b]
a < x ≤ b (a, b]
a ≤ x < b [a, b)
x ≥ a [a, ∞)
x > a (a, ∞)
x < a (−∞, a)
x ≤ a (−∞, a]
Working with Compound Inequalities
Compound inequalities connect two separate conditions using logical operators:
- "And" compounds (conjunction): Both conditions must be true simultaneously. The solution is the intersection of the two sets. For example, x > 1 AND x < 5 yields (1, 5).
- "Or" compounds (disjunction): At least one condition must be true. The solution is the union of the two sets, often requiring multiple interval notation segments. For instance, x < −2 OR x > 3 writes as (−∞, −2) ∪ (3, ∞).
When combining inequalities of the same direction (both > or both ≤), the stricter condition determines the result. When directions oppose, you typically obtain a bounded interval from the "and" case or a union of unbounded intervals from the "or" case.
Common Pitfalls and Best Practices
Avoid these frequent mistakes when converting between inequality and interval forms:
- Confusing Bracket and Parenthesis Symbols — Parentheses (a, b) always exclude endpoints and match strict inequalities (<, >). Square brackets [a, b] always include endpoints and match inclusive inequalities (≤, ≥). Reversing these symbols inverts your solution set, leading to incorrect answers.
- Forgetting the Union Symbol in "Or" Compounds — When inequalities are connected by "or" and the sets do not overlap, write the interval union explicitly: (−∞, 0) ∪ (5, ∞). Omitting the union symbol suggests a single connected interval, which is mathematically incorrect.
- Misidentifying Boundary Types — Double-check which inequality operator appears at each boundary. An inequality with a ≤ at the lower bound and < at the upper bound, like 2 ≤ x < 7, must be written as [2, 7)—mixing bracket types intentionally, not by mistake.
- Overlooking Infinity's Never-Closed Status — Infinity is never included in any interval; always use a parenthesis, never a bracket. Write (5, ∞) and (−∞, 3), never [5, ∞] or [−∞, 3]. Infinity is not a real number, only a direction.