Intervals are one of those mathematical ideas that look simple at first, but they appear everywhere: in inequalities, graphs, functions, domains, ranges, probability, and even real-life measurements such as temperature, time, and distance. Learning how to read, write, and solve interval problems gives you a strong foundation for algebra, pre-calculus, and calculus.
TLDR: Interval practice helps you understand how numbers are grouped on a number line using symbols like ( ), [ ], <, and ≤. Open intervals exclude endpoints, while closed intervals include them. The best way to master intervals is to translate between inequalities, number lines, and interval notation, then solve mixed practice questions step by step.
What Is an Interval?
An interval is a set of numbers between two values. For example, all numbers between 2 and 7 form an interval. Depending on whether 2 and 7 are included, we write the interval in different ways.
- Open interval: (2, 7) means all numbers between 2 and 7, but not 2 or 7.
- Closed interval: [2, 7] means all numbers from 2 to 7, including both endpoints.
- Half-open interval: [2, 7) means 2 is included, but 7 is not.
- Infinite interval: (-∞, 5] means all numbers less than or equal to 5.
The key rule is simple: parentheses mean “not included,” and brackets mean “included.” Infinity always uses parentheses because infinity is not an actual number you can reach.
Image not found in postmetaWhy Interval Notation Matters
Interval notation is a clean and compact way to describe many numbers at once. Instead of writing “all real numbers greater than 3 and less than or equal to 10,” we can simply write (3, 10]. This saves space and makes mathematical work easier to understand.
Intervals are especially useful when solving inequalities. A solution to an inequality is often not just one number; it is a whole range of numbers. For example, the inequality x > 4 has infinitely many solutions: 5, 6, 4.1, 100, and so on. In interval notation, the solution is (4, ∞).
Basic Interval Practice Questions With Solutions
Let’s begin with direct translation questions. These build confidence before moving into harder examples.
Question 1: Write the inequality as interval notation
Problem: Write x > 6 in interval notation.
Solution: The inequality says that x is greater than 6. Since 6 is not included, we use a parenthesis. The values continue forever to the right, so we use infinity.
Answer: (6, ∞)
Question 2: Write the interval as an inequality
Problem: Write [-3, 8) as an inequality.
Solution: The bracket at -3 means -3 is included. The parenthesis at 8 means 8 is not included. Therefore, x is at least -3 and less than 8.
Answer: -3 ≤ x < 8
Question 3: Identify the endpoints
Problem: What are the endpoints of (-5, 12], and which are included?
Solution: The endpoints are -5 and 12. The parenthesis beside -5 means -5 is not included. The bracket beside 12 means 12 is included.
Answer: Endpoint -5 is excluded; endpoint 12 is included.
Practice Questions Involving Number Lines
Number lines make intervals easier to visualize. An open circle means the endpoint is not included, while a closed circle means it is included. A shaded line shows all possible values.
Question 4: Convert a number line to interval notation
Problem: A number line has an open circle at 1, a closed circle at 9, and shading between them. Write the interval.
Solution: The open circle at 1 means 1 is not included. The closed circle at 9 means 9 is included. The shading between them means all numbers between 1 and 9 are part of the solution.
Answer: (1, 9]
Question 5: Draw the interval on a number line
Problem: Draw [-4, 3) on a number line.
Solution: Place a closed circle at -4 because the bracket includes -4. Place an open circle at 3 because the parenthesis excludes 3. Shade the line segment between -4 and 3.
Answer: Closed circle at -4, open circle at 3, shaded between them.
Image not found in postmetaSolving Inequalities and Writing Intervals
Many interval questions require solving an inequality first. Once the inequality is solved, the final answer can be written using interval notation.
Question 6: Solve a simple inequality
Problem: Solve 2x + 5 < 13 and write the answer in interval notation.
Solution:
- Subtract 5 from both sides: 2x < 8
- Divide both sides by 2: x < 4
- Write the solution as an interval.
Answer: (-∞, 4)
Question 7: Solve an inequality with a negative coefficient
Problem: Solve -3x ≥ 12 and write the answer in interval notation.
Solution: Divide both sides by -3. Remember, when dividing an inequality by a negative number, the inequality sign reverses.
x ≤ -4
Answer: (-∞, -4]
This is a common place where students make mistakes. The sign must flip from ≥ to ≤ because division by a negative changes the direction of the inequality.
Compound Inequalities and Intervals
A compound inequality combines two conditions. The word and usually means the overlap of two intervals, while or usually means the union of intervals.
Question 8: Solve a compound inequality
Problem: Solve -2 < x + 1 ≤ 6.
Solution: Subtract 1 from all three parts:
-3 < x ≤ 5
The left endpoint, -3, is not included. The right endpoint, 5, is included.
Answer: (-3, 5]
Question 9: Write a union of intervals
Problem: Write the solution to x < -1 or x ≥ 4 in interval notation.
Solution: The first condition, x < -1, becomes (-∞, -1). The second condition, x ≥ 4, becomes [4, ∞). Since the word is or, we join them with the union symbol.
Answer: (-∞, -1) ∪ [4, ∞)
Questions Involving Domain and Range
Intervals are often used to describe the domain and range of a function. The domain is the set of possible x-values, while the range is the set of possible y-values.
Question 10: Find the domain from a graph description
Problem: A graph begins at x = -2 with a closed point and continues to the right forever. Write the domain in interval notation.
Solution: Since the graph starts at -2 and includes that value, we use a bracket. Since it continues right forever, we use infinity with a parenthesis.
Answer: [-2, ∞)
Question 11: Find the range from a graph description
Problem: A graph has its lowest y-value at -6, but that point is not included. The graph rises without limit. Write the range.
Solution: The y-values are greater than -6, but not equal to -6. They continue upward forever.
Answer: (-6, ∞)
Challenging Interval Practice Questions
Now let’s look at slightly more advanced questions. These require careful thinking about overlap, restrictions, and endpoints.
Question 12: Find the intersection
Problem: Find the intersection of [1, 7] and (4, 10).
Solution: The intersection means the numbers that appear in both intervals. The first interval runs from 1 to 7, including both endpoints. The second interval runs from greater than 4 to less than 10. The overlap is greater than 4 and up to 7. Since 4 is not included, use a parenthesis. Since 7 is included in both intervals, use a bracket.
Answer: (4, 7]
Question 13: Find the union
Problem: Find the union of (-3, 2] and [2, 6).
Solution: The first interval ends at 2 and includes it. The second interval begins at 2 and also includes it. Together, they form one continuous interval from -3 to 6. The -3 is not included, and 6 is not included.
Answer: (-3, 6)
Question 14: Solve and combine interval notation
Problem: Solve |x – 2| < 5.
Solution: An absolute value inequality of the form |A| < b becomes -b < A < b.
-5 < x – 2 < 5
Add 2 to all parts:
-3 < x < 7
Answer: (-3, 7)
Common Mistakes to Avoid
Interval notation becomes much easier when you avoid a few predictable errors. Watch out for these:
- Using brackets with infinity: Never write [∞ or ∞]. Infinity always uses parentheses.
- Forgetting to flip the inequality sign: When multiplying or dividing by a negative number, reverse the sign.
- Confusing open and closed endpoints: Parentheses exclude endpoints; brackets include them.
- Mixing up union and intersection: Union combines intervals; intersection keeps only the overlap.
- Ignoring graph endpoints: Open circles and closed circles are important clues.
Extra Practice Set
Try these questions before checking the answers below.
- Write x ≤ 11 in interval notation.
- Write (0, 15] as an inequality.
- Solve 4x – 1 ≥ 19 and write the solution as an interval.
- Find the intersection of (-∞, 8] and [3, 12).
- Solve |x + 4| ≤ 2 and write the answer in interval notation.
Answers and Solutions
- Answer: (-∞, 11]. Since 11 is included, use a bracket.
- Answer: 0 < x ≤ 15. Zero is excluded, while 15 is included.
- Solution: Add 1 to both sides: 4x ≥ 20. Divide by 4: x ≥ 5. Answer: [5, ∞).
- Answer: [3, 8]. The overlap starts at 3 and ends at 8, and both endpoints are included.
- Solution: Rewrite as -2 ≤ x + 4 ≤ 2. Subtract 4 from all parts: -6 ≤ x ≤ -2. Answer: [-6, -2].
Final Thoughts
Interval notation is not just a shortcut; it is a powerful language for describing sets of numbers. Once you understand endpoints, inequalities, number lines, unions, and intersections, interval problems become much more predictable. The most effective way to improve is to practice translating between different forms: words, inequalities, graphs, and intervals.
As you work through interval practice questions, slow down and ask yourself: Are the endpoints included? Does the interval go left or right forever? Is this an overlap or a union? These small questions will help you avoid mistakes and build lasting confidence with intervals.
