Interval Notation: Complete Guide with Examples and Practice Problems
Interval notation is the standard mathematical shorthand for describing a range of real numbers on the number line — and once you understand the two symbols that drive it, the entire system clicks into place. You will see interval notation in algebra when solving inequalities, in pre-calculus when stating the domain and range of functions, and in calculus when specifying where a function is increasing, decreasing, or continuous. This guide covers every type of interval from the ground up, shows exactly how to convert any inequality into the correct notation, works through fully solved examples for domains and ranges, and ends with ten practice problems so you can check your skills before the next test.
Contents
- 01What Is Interval Notation?
- 02The Two Key Symbols: Parentheses vs. Brackets
- 03The Four Types of Intervals
- 04How to Write Interval Notation from an Inequality
- 05Worked Examples: Converting Single Inequalities
- 06Compound Inequalities and Interval Notation
- 07Union and Intersection of Intervals
- 08Interval Notation for Domain and Range
- 09Common Mistakes with Interval Notation
- 10Absolute Value Inequalities and Interval Notation
- 11Practice Problems with Full Solutions
- 12FAQ: Interval Notation Questions Answered
What Is Interval Notation?
Interval notation is a concise way of representing a continuous set of real numbers between two boundary values. Instead of writing out the full inequality −3 < x ≤ 7, you write (−3, 7]. The notation tells a reader immediately whether each boundary is included or excluded and whether the set extends to infinity. Mathematicians, textbooks, and standardized tests use interval notation because it is faster to write and unambiguous — one glance tells you everything about the solution set. You will encounter interval notation on the SAT, ACT, and every college-level math course. It also appears in textbook answers for domain and range, in calculus for intervals of increase and concavity, and anywhere a solution spans a continuous range of values.
Interval notation uses parentheses () for excluded endpoints and brackets [] for included endpoints. Infinity always takes a parenthesis — it is never reached, so it can never be included.
The Two Key Symbols: Parentheses vs. Brackets
The entire system of interval notation rests on two symbols and one rule about infinity. A parenthesis ( or ) means the endpoint next to it is NOT included in the set — the interval is open at that end. A bracket [ or ] means the endpoint IS included — the interval is closed at that end. Infinity (∞) and negative infinity (−∞) always appear with parentheses, because infinity is a concept, not a number you can actually reach. Mixing up parentheses and brackets is the single most common source of wrong answers, so take time now to make the distinction automatic.
1. Parenthesis ( or ): endpoint is excluded
Use a parenthesis when the boundary value does NOT satisfy the original inequality. If the inequality uses strict < or >, the endpoint is excluded. Example: x > 4 gives (4, ∞) — the value 4 is not in the solution because 4 is not greater than 4.
2. Bracket [ or ]: endpoint is included
Use a bracket when the boundary value DOES satisfy the inequality. If the inequality uses ≤ or ≥, the endpoint is included. Example: x ≥ 4 gives [4, ∞) — the value 4 is in the solution because 4 ≥ 4 is true.
3. Infinity always uses parentheses
Whether you write (−∞, 5) or (0, ∞), the infinity side always gets a parenthesis. Writing [∞] is a notation error. All real numbers — the entire number line — is written as (−∞, ∞).
The Four Types of Intervals
Every set you will encounter in algebra and pre-calculus fits into one of four interval types. Recognizing each type makes converting between inequalities and interval notation automatic rather than something you need to puzzle through each time.
1. Open interval (a, b): neither endpoint included
Parentheses on both sides. Inequality equivalent: a < x < b. Example: (2, 9) means all real numbers strictly between 2 and 9. Neither 2 nor 9 belongs to the set. On a number line, open circles appear at both 2 and 9.
2. Closed interval [a, b]: both endpoints included
Brackets on both sides. Inequality equivalent: a ≤ x ≤ b. Example: [−5, 3] means all real numbers from −5 to 3, including both endpoints. On a number line, filled circles appear at both −5 and 3.
3. Half-open interval [a, b) or (a, b]: one included, one excluded
[a, b) means a ≤ x < b — left endpoint included, right excluded. (a, b] means a < x ≤ b — right endpoint included, left excluded. Example: [0, 5) covers all numbers from 0 up to but not including 5. It includes 0, 2.7, 4.999, but not 5.
4. Unbounded intervals: extending to infinity
(a, ∞) means x > a. [a, ∞) means x ≥ a. (−∞, b) means x < b. (−∞, b] means x ≤ b. (−∞, ∞) is the entire real number line — every real number. Unbounded intervals always pair infinity with a parenthesis.
Open: neither endpoint included. Closed: both included. Half-open: one included, one excluded. Unbounded: extends to ∞ or −∞ on at least one side.
How to Write Interval Notation from an Inequality
Converting between an inequality and interval notation follows a direct, step-by-step process. Once you practice this procedure a few times, it becomes second nature on any test or homework assignment.
1. Step 1: Identify the boundary values
Find the numbers (or expressions) that x is being compared against. For x > −3, the boundary is −3. For −1 < x ≤ 8, the boundaries are −1 (left) and 8 (right).
2. Step 2: Assign a symbol to each endpoint
If the inequality at a boundary is strict (< or >), use a parenthesis at that end. If the inequality includes equality (≤ or ≥), use a bracket. Infinity always gets a parenthesis regardless.
3. Step 3: Write the interval from left to right
Intervals are always written smaller value on the left, larger on the right. Write: left symbol, left boundary, comma, right boundary, right symbol. For −1 < x ≤ 8: left is −1 with <, so parenthesis; right is 8 with ≤, so bracket. Answer: (−1, 8].
4. Step 4: Handle unbounded inequalities with ∞
If the set extends infinitely in one direction, use −∞ or ∞ as that boundary with a parenthesis. x > 5 becomes (5, ∞). x ≤ −2 becomes (−∞, −2].
5. Step 5: Verify with a test value
Pick a number inside your interval and confirm it satisfies the original inequality. Pick a number outside and confirm it does not. This 30-second check catches parenthesis/bracket errors before they cost you points.
Worked Examples: Converting Single Inequalities
These eight examples cover every standard case that appears on homework and tests. Each one applies the five-step process above. Work through the first few before reading the solution.
1. Example 1: x > 3
Boundary 3, strict >: parenthesis. Extends right to ∞: parenthesis. Answer: (3, ∞). Check: x = 10 satisfies 10 > 3 ✓. x = 1 does not satisfy 1 > 3 ✓.
2. Example 2: x ≥ −7
Boundary −7, non-strict ≥: bracket. Extends right to ∞: parenthesis. Answer: [−7, ∞). Check: x = −7 satisfies −7 ≥ −7 ✓. x = −10 does not satisfy −10 ≥ −7 ✓.
3. Example 3: x < 2
Boundary 2, strict <: parenthesis. Extends left to −∞: parenthesis. Answer: (−∞, 2). Check: x = 0 satisfies 0 < 2 ✓. x = 5 does not satisfy 5 < 2 ✓.
4. Example 4: x ≤ 0
Boundary 0, non-strict ≤: bracket. Extends left to −∞: parenthesis. Answer: (−∞, 0]. Check: x = 0 satisfies 0 ≤ 0 ✓. x = 1 does not satisfy 1 ≤ 0 ✓.
5. Example 5: −4 < x < 6
Left boundary −4, strict <: parenthesis. Right boundary 6, strict <: parenthesis. Answer: (−4, 6). Check: x = 0 satisfies −4 < 0 < 6 ✓. x = 6 fails at 6 < 6 ✓.
6. Example 6: −3 ≤ x < 10
Left boundary −3, non-strict ≤: bracket. Right boundary 10, strict <: parenthesis. Answer: [−3, 10). Check: x = −3 satisfies −3 ≤ −3 < 10 ✓. x = 10 fails at 10 < 10 ✓.
7. Example 7: −2 ≤ x ≤ 5
Both boundaries are non-strict: brackets on both sides. Answer: [−2, 5]. Check: x = −2 satisfies −2 ≤ −2 ≤ 5 ✓. x = 6 does not satisfy 6 ≤ 5 ✓.
8. Example 8: All real numbers except x = 4
Remove a single point: split the line into two pieces. Answer: (−∞, 4) ∪ (4, ∞). This pattern arises constantly in rational function domains where a single x-value makes the denominator zero.
Conversion rule: ≤ or ≥ → bracket [ or ]. Strict < or > → parenthesis ( or ). Infinity always → parenthesis.
Compound Inequalities and Interval Notation
Compound inequalities connect two conditions with 'and' or 'or'. These translate directly into interval notation — 'and' produces a single bounded interval (the two conditions must overlap), while 'or' produces two separate intervals joined by the union symbol ∪. Understanding this distinction prevents the most common compound-inequality error: using one interval where two are needed (or vice versa).
1. Compound 'and': −2 ≤ x ≤ 5
Both conditions hold simultaneously. Left side ≤: bracket. Right side ≤: bracket. Answer: [−2, 5]. All numbers from −2 to 5, including both endpoints.
2. Compound 'and' with mixed signs: 0 < x ≤ 12
Left side strict <: parenthesis. Right side non-strict ≤: bracket. Answer: (0, 12]. Numbers greater than 0 and at most 12. Check: x = 0 fails (0 < 0 is false) ✓. x = 12 passes (0 < 12 ≤ 12) ✓.
3. Compound 'or': x < −1 or x ≥ 4
Each condition gives its own interval. x < −1 → (−∞, −1). x ≥ 4 → [4, ∞). Join with ∪: (−∞, −1) ∪ [4, ∞). This set has a gap — numbers between −1 and 4 satisfy neither condition.
4. Solve first, then convert: −5 < 2x + 1 ≤ 9
Subtract 1 from all three parts: −6 < 2x ≤ 8. Divide by 2 (positive — no flip): −3 < x ≤ 4. Answer: (−3, 4]. Always finish solving the inequality before translating.
5. Solve first, then convert: 3x − 6 > 9 or 2x + 1 < −3
Solve each: 3x > 15 → x > 5, giving (5, ∞). And 2x < −4 → x < −2, giving (−∞, −2). Since 'or', join: (−∞, −2) ∪ (5, ∞).
'And' compound inequalities → one interval. 'Or' compound inequalities → two intervals joined by ∪.
Union and Intersection of Intervals
When absolute value inequalities and quadratic inequalities produce multi-piece solutions, you need to combine intervals using union (∪) or intersection (∩). Union means 'or': a number belongs to the combined set if it is in at least one interval. Intersection means 'and': a number belongs only if it is in both intervals at the same time. These operations appear in pre-calculus domain problems, in set theory, and in calculus when describing positive or negative regions of a function.
1. Union example: (−∞, 2) ∪ (5, ∞)
This means x < 2 OR x > 5. Numbers between 2 and 5 (including 2 and 5 themselves) are NOT in the set. On a number line, shade left of 2 with an open circle and right of 5 with an open circle. Typical result for |x − 3.5| > 1.5.
2. Union example: (−∞, −3] ∪ [1, ∞)
This means x ≤ −3 OR x ≥ 1. Both −3 and 1 are included (brackets). Numbers strictly between −3 and 1 are excluded. Typical result for an absolute value inequality like |x + 1| ≥ 2.
3. Intersection example: [−4, 6] ∩ [0, 10]
Find the overlap. The left boundary of the overlap is max(−4, 0) = 0. The right boundary is min(6, 10) = 6. Since both 0 and 6 are closed (bracketed) in their respective intervals, keep brackets. Answer: [0, 6].
4. Intersection example: (1, 8) ∩ [5, 12)
Left boundary: max(1, 5) = 5. In (1, 8), the value 5 is an interior point, so no exclusion there. In [5, 12), 5 is the left endpoint with a bracket — included. Use bracket for 5. Right boundary: min(8, 12) = 8. In (1, 8), 8 is excluded by its parenthesis. Answer: [5, 8).
Intersection: left boundary = larger of the two left endpoints; right boundary = smaller of the two right endpoints. Inherit the stricter symbol (parenthesis beats bracket) at each boundary.
Interval Notation for Domain and Range
Domain and range are the most frequent real-world applications of interval notation in pre-calculus. The domain is all valid x-values (inputs), and the range is all achievable y-values (outputs). Interval notation expresses both cleanly and precisely. The strategy for domain is always: identify what would break the function (division by zero, square root of a negative, logarithm of a non-positive number) and exclude those values. For range, determine the minimum or maximum output and identify any gaps.
1. Linear function: f(x) = 2x − 5
No restrictions on input or output. Domain: (−∞, ∞). Range: (−∞, ∞). Every real number can be plugged in, and every real number appears as an output.
2. Square root function: f(x) = √(x − 4)
Require x − 4 ≥ 0 → x ≥ 4. Domain: [4, ∞). The output √(x − 4) is always ≥ 0, and f(4) = 0 is achievable. Range: [0, ∞). Note the bracket at 4 because f(4) = √0 = 0 — the endpoint is reached.
3. Rational function: f(x) = 3/(x − 5)
Denominator cannot equal zero: x ≠ 5. Domain: (−∞, 5) ∪ (5, ∞). The function approaches but never reaches y = 0 (horizontal asymptote). Range: (−∞, 0) ∪ (0, ∞).
4. Quadratic function: f(x) = x² − 6x + 5 (upward parabola)
Domain: (−∞, ∞) — all inputs valid. Vertex x = −b/(2a) = 6/2 = 3. Minimum output: f(3) = 9 − 18 + 5 = −4. Since the parabola opens upward, every y-value ≥ −4 is achievable. Range: [−4, ∞).
5. Logarithmic function: f(x) = ln(2x + 6)
Argument must be positive: 2x + 6 > 0 → 2x > −6 → x > −3. Domain: (−3, ∞). Parenthesis at −3 because the inequality is strict. The logarithm can output any real number. Range: (−∞, ∞).
6. Rational function with two excluded points: g(x) = 1/(x² − 9)
x² − 9 = 0 → x = 3 or x = −3. Both are excluded. Domain: (−∞, −3) ∪ (−3, 3) ∪ (3, ∞). Three separate pieces joined by ∪.
For domain: exclude x-values that cause division by zero, square root of a negative, or log of a non-positive number. For range: find the vertex or asymptote that caps or floors the output.
Common Mistakes with Interval Notation
Most errors with interval notation fall into a small number of predictable patterns. Spotting these before you make them is far more efficient than learning from lost points on a test.
1. Putting a bracket next to infinity
Writing [3, ∞] or [−∞, 5] is always wrong. Infinity is a concept, not a reachable number, so it can never be included. Correct forms: [3, ∞) and (−∞, 5].
2. Swapping brackets and parentheses
The pattern is: ≤ and ≥ (equality included) → brackets [ ]. Strict < and > (equality excluded) → parentheses ( ). A quick mnemonic: the bracket 'grabs' the number, just like ≤ 'grabs' the boundary value into the solution.
3. Writing the interval in reverse order
Intervals always go smaller to larger, left to right. Writing (8, 3) is wrong — that represents the empty set in standard notation. If your solution is −5 < x < 2, write (−5, 2), not (2, −5).
4. Forgetting to solve the inequality before converting
Translating −6 < 3x ≤ 12 directly without solving first is a common shortcut that causes errors. Divide by 3 first: −2 < x ≤ 4. Then convert: (−2, 4]. Always simplify completely before writing the interval.
5. Using a single interval for an 'or' compound solution
The solution to x < −2 or x > 7 is NOT (−2, 7) — that would mean −2 < x < 7, which is the opposite of what you want. The correct answer is (−∞, −2) ∪ (7, ∞). Any solution with a gap requires two intervals connected by ∪.
6. Using ∪ for an 'and' compound inequality
Conversely, −3 < x AND x ≤ 8 simplifies to −3 < x ≤ 8, which is one interval: (−3, 8]. Writing this as (−∞, 8] ∪ (−3, ∞) is wrong — that union would include numbers outside the intended range.
Absolute Value Inequalities and Interval Notation
Absolute value inequalities are one of the most common sources of multi-interval solutions. The two standard forms each produce a predictable structure that you can write in interval notation once you know the pattern.
1. Case 1: |x − a| < r (less-than type) → single interval
The solution is always a single interval centered at a with radius r. Rewrite as −r < x − a < r, then add a to all three parts: a − r < x < a + r. Answer: (a − r, a + r). Example: |x − 3| < 5 → −5 < x − 3 < 5 → −2 < x < 8 → (−2, 8).
2. Case 2: |x − a| > r (greater-than type) → two intervals
The solution is two pieces going away from the center. Rewrite as x − a < −r OR x − a > r, giving x < a − r or x > a + r. Answer: (−∞, a − r) ∪ (a + r, ∞). Example: |x − 3| > 5 → x < −2 or x > 8 → (−∞, −2) ∪ (8, ∞).
3. With ≤ and ≥: |x + 2| ≤ 4
Non-strict, so use brackets at the boundaries. −4 ≤ x + 2 ≤ 4. Subtract 2: −6 ≤ x ≤ 2. Answer: [−6, 2]. Check: x = −6 gives |−6 + 2| = |−4| = 4 ≤ 4 ✓.
4. With ≥: |2x − 1| ≥ 7
Non-strict on a greater-than-type: use brackets at the boundaries. 2x − 1 ≤ −7 OR 2x − 1 ≥ 7. Left: 2x ≤ −6 → x ≤ −3. Right: 2x ≥ 8 → x ≥ 4. Answer: (−∞, −3] ∪ [4, ∞).
|x − a| < r gives one interval (a − r, a + r). |x − a| > r gives two intervals: (−∞, a − r) ∪ (a + r, ∞). Swap to brackets when the inequality is ≤ or ≥.
Practice Problems with Full Solutions
Work through all ten problems before reading the solutions. They progress from basic single-inequality conversion through compound, union, domain, and quadratic problems. If you can solve all ten, your skills are ready for the next exam.
1. Problem 1: Write x > −6 using interval notation
Strict >, so parenthesis at −6. Extends right to ∞: parenthesis. Answer: (−6, ∞).
2. Problem 2: Write x ≤ 4 using interval notation
Non-strict ≤, so bracket at 4. Extends left to −∞: parenthesis. Answer: (−∞, 4].
3. Problem 3: Write −5 ≤ x < 3 using interval notation
Left boundary −5 with ≤: bracket. Right boundary 3 with <: parenthesis. Answer: [−5, 3).
4. Problem 4: Solve 3x − 9 > 0, then write in interval notation
3x > 9 → x > 3. Strict >, parenthesis at 3. Answer: (3, ∞).
5. Problem 5: Solve −4 ≤ 2x + 2 < 8, then convert
Subtract 2 from all parts: −6 ≤ 2x < 6. Divide by 2: −3 ≤ x < 3. Left boundary −3 with ≤: bracket. Right boundary 3 with <: parenthesis. Answer: [−3, 3).
6. Problem 6: Write x ≤ 0 or x > 5 in interval notation
x ≤ 0 → (−∞, 0]. x > 5 → (5, ∞). Join: (−∞, 0] ∪ (5, ∞).
7. Problem 7: Find [−3, 5] ∩ [1, 8]
Overlap left = max(−3, 1) = 1 (bracket from second interval; 1 is interior to first, so bracket). Overlap right = min(5, 8) = 5 (bracket from first interval; 5 is interior to second, so bracket). Answer: [1, 5].
8. Problem 8: Find the domain of f(x) = √(2x − 8)
Require 2x − 8 ≥ 0 → x ≥ 4. Non-strict, so bracket. Answer: [4, ∞).
9. Problem 9: Find the domain of g(x) = 5/(x² − 9)
x² − 9 ≠ 0 → x ≠ 3 and x ≠ −3. Remove both points from the real line. Answer: (−∞, −3) ∪ (−3, 3) ∪ (3, ∞).
10. Problem 10: Find the range of h(x) = −x² + 4 on x ∈ [−2, 2]
Downward parabola. Vertex at x = 0: h(0) = 4 (maximum). At endpoints: h(±2) = −4 + 4 = 0 (minimum on this domain). Range runs from 0 up to 4, both included. Answer: [0, 4].
FAQ: Interval Notation Questions Answered
Here are the questions students most commonly ask when learning interval notation for the first time.
1. Why use interval notation instead of just writing inequalities?
Both describe the same set, but interval notation is the standard in higher-level mathematics. Textbooks, solutions manuals, calculators, and standardized test answer keys all use it. Learning it now prevents confusion in pre-calculus, calculus, and analysis courses.
2. Can both endpoints of an interval be the same number?
[a, a] is a valid interval — it contains exactly one point, a. The open interval (a, a) contains no elements and represents the empty set ∅. These degenerate cases appear when a domain restriction collapses to a single point.
3. How do I tell an interval from a coordinate pair like (3, 7)?
Context is key. In any problem involving a single variable inequality, domain, or solution set, (3, 7) is an interval meaning 3 < x < 7. In a two-variable geometry context, (3, 7) is the point x = 3, y = 7. If the problem is about a number line or a function's domain, it is an interval.
4. What does it mean when interval notation shows three pieces like (−∞, −3) ∪ (−3, 3) ∪ (3, ∞)?
This means all real numbers except −3 and 3. Each ∪ joins the pieces, and the two gaps at −3 and 3 indicate those points are excluded. This pattern is exactly the domain of a rational function where two x-values make the denominator zero.
5. Is (−∞, ∞) the same as writing ℝ?
Yes. ℝ (the set of all real numbers) and (−∞, ∞) mean the same thing. ℝ is shorthand; (−∞, ∞) is the explicit interval notation form. Either is accepted on most courses, but using (−∞, ∞) is clearer on a test when interval notation is explicitly requested.
6. Does interval notation work for integers only, or all real numbers?
Interval notation describes continuous sets of real numbers — not just integers. The interval (1, 5) includes 1.5, 2.7, π, √3, and infinitely many other values between 1 and 5. If a problem restricts to integers, it will say so explicitly (using set notation like {2, 3, 4}).
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