ACT Math: Use Set Notation and Interval Notation to Express Solutions Correctly

Set notation uses braces: {x | x>5} means "the set of all x such that x is greater than 5." Interval notation uses parentheses and brackets: (5, ∞) means the same thing (numbers greater than 5, not including 5). Use parentheses ( or ) for open intervals (not including the endpoint); use brackets [ or ] for closed intervals (including the endpoint). Examples: [3, 7] includes 3 and 7; (3, 7) excludes both; [3, 7) includes 3 but not 7. ACT Math asks you to write solutions using one of these notations, and getting the notation right is essential for getting the question right.

Example: Solve x+2>5. Solution: x>3. Set notation: {x | x>3}. Interval notation: (3, ∞). Both are correct; the question will specify which notation to use. Another example: Solve 1≤x<6. Interval notation: [1, 6) (includes 1, excludes 6). Set notation: {x | 1≤x<6}. Notice the subtle difference: the inequality uses ≤ (includes 1) and < (excludes 6), matching the bracket [ and parenthesis ).

Trap 1: Confusing parentheses and brackets. (5, 10) excludes 5 and 10; [5, 10] includes both. If the solution is x≥5 and x<10, the interval is [5, 10) (bracket for ≥, parenthesis for <). Trap 2: Using infinity incorrectly. Infinity is never included, so always use parentheses: (-∞, 5) or (3, ∞). Never write [∞) or (-∞]. When you write an interval, check: Does my bracket notation match my inequality signs? Are my endpoints correct? Is infinity written with a parenthesis, not a bracket?

Before you lock in an interval notation answer, re-read the inequality and verify that your brackets and parentheses match the inequality signs (≥ and ≤ use brackets; > and < use parentheses).

Inequality 1: x>-2. Interval: (-2, ∞). Set: {x | x>-2}. Inequality 2: -5≤x≤3. Interval: [-5, 3]. Set: {x | -5≤x≤3}. Inequality 3: x<0 or x>10. Interval: (-∞, 0)∪(10, ∞) (union symbol ∪ means "or"). Set: {x | x<0 or x>10}. Inequality 4: 1Notice how the brackets and parentheses in interval notation directly mirror the inequality signs: < and > use parentheses; ≤ and ≥ use brackets.

Do ten more inequality-to-interval conversions daily for one week. By test day, you'll convert notation so fast that you won't even consciously think about it.

ACT Math often asks you to express solutions in specific notation (interval, set, or inequality form). Getting the notation right is part of getting the question right. Students who know both notations and can convert between them will never lose points to notation confusion, gaining a small but consistent advantage.

Learn interval and set notation this week. Drill conversion from inequalities to both notations. By test day, notation will feel natural and automatic, freeing your mental energy for harder problem-solving steps.