Example 1: Converting from Inequality

Problem: Express -2 ≤ x < 5 in interval notation.

Solution: The inequality includes -2 (≤ sign) but excludes 5 (< sign).
Interval notation: [-2, 5)

The bracket at -2 shows it's included; the parenthesis at 5 shows it's excluded.

Example 2: Infinite Interval

Problem: Express x ≥ 10 in interval notation.

Solution: This includes 10 and everything greater, extending to infinity.
Interval notation: [10, ∞)

The bracket at 10 shows it's included. We always use a parenthesis with infinity.

Example 3: Union of Intervals

Problem: Express x < -1 OR x > 3 in interval notation.

Solution: This represents two separate regions that don't touch.
First part: x < -1 is (-∞, -1)
Second part: x > 3 is (3, ∞)
Combined with union: (-∞, -1) ∪ (3, ∞)

Example 4: All Real Numbers Except One

Problem: Express all real numbers except x = 2.

Solution: We need everything before 2 and everything after 2, but not 2 itself.
Interval notation: (-∞, 2) ∪ (2, ∞)

This shows the "hole" at x = 2, common when dealing with function domains that have removable discontinuities or division by zero issues.

Match Inequality Symbols to Brackets

The inequality symbols directly tell you which bracket to use: ≤ or ≥ means use a bracket [ or ], while < or > means use a parenthesis ( or ). For example, x ≥ 5 becomes [5, ∞), while x > 5 becomes (5, ∞). This one-to-one correspondence makes conversion straightforward once you remember the pattern.

Always Use Parentheses with Infinity

Infinity (∞) is not a number—it's a concept representing unbounded growth. You can never "reach" infinity, so it can never be included in a set. Always write (∞) or (-∞), never [∞] or [-∞]. This is a strict convention in mathematics that you should never violate.

Write Numbers in Increasing Order

In an interval [a, b], always write the smaller number first: a < b. Writing [5, 2] is incorrect—it should be [2, 5]. For unions, list intervals from left to right on the number line: (-∞, -1) ∪ (3, ∞) is correct, while (3, ∞) ∪ (-∞, -1) looks disorganized (though technically equivalent).

Visualize with a Number Line

When in doubt, sketch a quick number line. Shade the regions that satisfy the inequality or belong to the set. Filled dots represent included endpoints (use brackets), open dots represent excluded endpoints (use parentheses). This visual approach helps prevent errors when converting between notations.

Interval

A continuous set of real numbers between (and possibly including) two endpoints. Intervals can be finite (bounded on both sides) or infinite (extending indefinitely in one or both directions). They represent all numbers in a range without gaps.

Closed Interval

An interval [a, b] that includes both endpoints a and b. Represented by square brackets. For example, [2, 5] includes 2, 5, and all numbers in between. Corresponds to the compound inequality a ≤ x ≤ b.

Open Interval

An interval (a, b) that excludes both endpoints a and b. Represented by parentheses. For example, (2, 5) includes all numbers between 2 and 5 but not 2 or 5 themselves. Corresponds to the compound inequality a < x < b.

Half-Open Interval

An interval that includes one endpoint but not the other, written as [a, b) or (a, b]. Also called a half-closed interval. For example, [2, 5) includes 2 but not 5, corresponding to 2 ≤ x < 5.

Union

The symbol ∪ means "or" and combines sets. For intervals, A ∪ B contains all numbers that are in A or in B (or in both). For example, [1, 3] ∪ [5, 7] includes numbers from 1 to 3 and separately from 5 to 7.

Intersection

The symbol ∩ means "and" and finds the overlap between sets. For intervals, A ∩ B contains only numbers that are in both A and B. For example, [1, 5] ∩ [3, 7] = [3, 5], the overlapping portion.

Endpoint

The boundary values a and b in an interval [a, b] or (a, b). Endpoints define where an interval starts and ends. Whether endpoints are included or excluded is indicated by the type of bracket used.