Understanding Interval Notation
Interval notation represents subsets of the real number line by specifying a lower and upper bound. The notation concisely captures whether endpoints are part of the set or merely boundaries.
- Closed interval
[a, b]includes both endpoints, representing all x where a ≤ x ≤ b. - Open interval
(a, b)excludes both endpoints, representing all x where a < x < b. - Half-open intervals
[a, b)or(a, b]include exactly one endpoint. - Unbounded intervals use the infinity symbol
∞with a parenthesis (never a bracket), such as[a, ∞)or(−∞, b].
The bracket–parenthesis distinction is critical: square brackets mean inclusion; parentheses mean exclusion.
Converting Inequalities to Interval Form
Start by identifying the variable's range from the inequality statement. For compound inequalities like a < x ≤ b, treat the left and right bounds separately.
- Write the lower bound first, followed by a comma and the upper bound.
- If the lower bound uses ≤ or ≥, place a square bracket
[before it; if < or >, use a parenthesis(. - Mirror this logic for the upper bound, placing
]or)after it. - For unbounded ranges approaching infinity, always use parentheses around ∞.
Example: 2 ≤ x < 7 becomes [2, 7), since 2 is included but 7 is not.
Bracket and Parenthesis Rules
The choice of bracket type depends on whether each endpoint is included in the solution set:
Closed: [a, b] means a ≤ x ≤ b (both endpoints included)
Open: (a, b) means a < x < b (both endpoints excluded)
Half-open left: [a, b) means a ≤ x < b
Half-open right: (a, b] means a < x ≤ b
Unbounded: (−∞, b] or [a, ∞) for infinite ranges
a— The lower bound (infimum) of the intervalb— The upper bound (supremum) of the intervalx— Any real number within the specified range
Special Cases and All Real Numbers
When a solution includes all real numbers, write (−∞, ∞). Both infinity symbols always pair with parentheses, never brackets, because infinity is not a real number that can be reached or included.
Compound inequalities using 'and' (both conditions true) narrow the range; those using 'or' (either condition true) may split into multiple disjoint intervals connected by the union symbol ∪. For instance, x < −1 or x ≥ 3 becomes (−∞, −1) ∪ [3, ∞).
Pay close attention to strict versus non-strict inequalities: a single < or ≤ difference determines whether an endpoint belongs to the set.
Common Pitfalls When Writing Interval Notation
Accuracy in interval notation hinges on careful attention to endpoint inclusion and proper use of brackets.
- Confusing bracket orientation — Square brackets always denote inclusion; parentheses always denote exclusion. Visualize a closed door (bracket) as 'included' and an open doorway (parenthesis) as 'excluded.' Reversing them fundamentally changes the meaning of your solution.
- Forgetting parentheses around infinity — Infinity is not a real number and can never be included in a set. Always wrap ±∞ in parentheses, never square brackets. <code>[5, ∞)</code> is correct; <code>[5, ∞]</code> is mathematically invalid.
- Mishandling compound inequalities with 'or' — When two conditions are joined by 'or,' the solution set splits into separate intervals. Use the union operator ∪ to connect them. For <em>x</em> < 2 or <em>x</em> ≥ 5, write <code>(−∞, 2) ∪ [5, ∞)</code>, not a single interval.