Interval Notation Calculator

What is interval notation?

Interval notation is a compact mathematical way to describe a continuous set of real numbers. Instead of listing every number in a solution set, which is usually impossible, interval notation gives the left boundary, the right boundary, and the endpoint rules. For example, the inequality \(2 < x < 7\) contains every real number greater than \(2\) and less than \(7\). Its interval notation is \((2,7)\). The parentheses show that \(2\) and \(7\) are not included. The numbers between them, such as \(2.1\), \(3\), \(4.5\), and \(6.999\), are included.

The notation is especially useful in algebra, precalculus, calculus, set theory, statistics, and graph analysis. When you solve an inequality, find a domain, describe where a function is increasing, identify a range, or write the answer to an absolute value inequality, interval notation is often the cleanest final form. It allows students, teachers, and exam graders to understand the solution set immediately without reading a long sentence.

An interval usually has the form \((a,b)\), \([a,b]\), \((a,b]\), or \([a,b)\). The symbols \(a\) and \(b\) are endpoints. The left endpoint is the smaller value, and the right endpoint is the larger value. If an endpoint is included, the interval uses a bracket. If an endpoint is excluded, the interval uses a parenthesis. This endpoint choice is the part that students most often get wrong, so the calculator above focuses strongly on endpoint inclusion.

These formulas show the direct relationship between inequality notation, set-builder notation, and interval notation. The open interval \((a,b)\) means both endpoints are excluded. The closed interval \([a,b]\) means both endpoints are included. The half-open interval \((a,b]\) excludes the left endpoint but includes the right endpoint. The half-open interval \([a,b)\) includes the left endpoint but excludes the right endpoint. The calculator uses the same rules when it converts your input.

How to use the interval notation calculator

The calculator has three modes. The first mode, inequality to interval notation, is best when the problem is written as an inequality. For example, if your worksheet says \( -3 \le x < 8 \), choose the matching inequality type, enter \(a=-3\), enter \(b=8\), and press the conversion button. The result will be \([-3,8)\). The bracket beside \(-3\) appears because the inequality includes equality at the left endpoint. The parenthesis beside \(8\) appears because \(x\) must be less than \(8\), not equal to \(8\).

The second mode, interval builder, is best when you already know the endpoints. Enter the lower endpoint, the upper endpoint, and choose whether each endpoint should be included. This is helpful for writing domains and ranges from graphs. If a graph starts at \(x=-1\) with a filled dot and ends at \(x=5\) with an open circle, you would enter lower endpoint \(-1\), upper endpoint \(5\), include the lower endpoint, and exclude the upper endpoint. The result is \([-1,5)\).

The third mode, union builder, lets you combine several separate intervals. This matters because not every solution set is one continuous piece. For example, the inequality \(x < -2\) or \(x \ge 5\) has two separate regions on the number line. It is written as \((-\infty,-2) \cup [5,\infty)\). The union symbol \( \cup \) means “or” in this context: the solution can be in the first interval or in the second interval.

Interval notation symbols and meanings

Each symbol in interval notation has a precise meaning. A parenthesis means “do not include this endpoint.” A bracket means “include this endpoint.” The comma separates the left boundary from the right boundary. The infinity symbol \( \infty \) means the interval continues forever to the right, and \( -\infty \) means the interval continues forever to the left. Infinity is not a real number, so it can never be included as an endpoint. That is why intervals such as \([2,\infty]\) and \([- \infty,5]\) are incorrect. The correct forms are \([2,\infty)\) and \((-\infty,5]\).

The union symbol \( \cup \) joins two or more intervals into one solution set. For example, \((-\infty,1) \cup (4,\infty)\) means all real numbers less than \(1\), together with all real numbers greater than \(4\). The numbers between \(1\) and \(4\) are not included. This kind of answer appears often when solving quadratic inequalities, rational inequalities, and absolute value inequalities.

Set-builder notation describes the same idea in a more verbal mathematical format. For example, \(\{x \in \mathbb{R}\mid x \ge 3\}\) means “the set of all real numbers \(x\) such that \(x\) is greater than or equal to \(3\).” The matching interval notation is \([3,\infty)\). Interval notation is usually shorter, while set-builder notation can be more explicit. Students should understand both because many textbooks, exams, and graphing problems switch between them.

How inequalities convert to interval notation

The fastest way to convert an inequality into interval notation is to read the inequality from left to right. In a bounded inequality such as \(1 < x \le 9\), the left boundary is \(1\), the variable is in the middle, and the right boundary is \(9\). Since \(1 < x\) does not include \(1\), the interval starts with a parenthesis. Since \(x \le 9\) includes \(9\), the interval ends with a bracket. The answer is \((1,9]\).

For a one-sided inequality, one endpoint is finite and the other endpoint is infinite. The inequality \(x > 4\) means the values continue forever to the right, so the interval is \((4,\infty)\). The endpoint \(4\) is excluded because the sign is \(>\), not \( \ge \). The inequality \(x \ge 4\) becomes \([4,\infty)\). The bracket appears because \(4\) is included. Notice that the infinity side still uses a parenthesis.

For inequalities that point to the left, the interval begins with negative infinity. The inequality \(x < 6\) becomes \((-\infty,6)\), while \(x \le 6\) becomes \((-\infty,6]\). The right endpoint changes from a parenthesis to a bracket when equality is included, but the negative infinity side remains a parenthesis. This is one of the most important fixed rules in interval notation.

Compound inequalities with “or” usually require a union. For example, \(x < -3\) or \(x \ge 2\) becomes \((-\infty,-3) \cup [2,\infty)\). Compound inequalities with “and” often become one bounded interval. For example, \(x > -3\) and \(x \le 2\) becomes \((-3,2]\). When translating a word problem, pay close attention to whether the condition is an “and” condition or an “or” condition, because that determines whether you need one interval or a union of intervals.

Worked examples

Example 1: Convert a bounded inequality

Convert \( -4 < x \le 10 \) into interval notation.

The answer is \((-4,10]\). This means every real number greater than \(-4\) and less than or equal to \(10\). The value \(-4\) is not allowed, but \(10\) is allowed.

Example 2: Convert a one-sided inequality

Convert \(x \ge 12\) into interval notation.

The answer is \([12,\infty)\). This includes \(12\), \(13\), \(100\), and every larger real number. It does not have a largest value.

Example 3: Convert a union inequality

Convert \(x < -2\) or \(x > 5\) into interval notation.

The answer has two separate pieces because the values between \(-2\) and \(5\) are not included. This type of interval often appears when a quadratic inequality is positive outside its roots.

Example 4: Domain of a square root function

Find the domain of \(f(x)=\sqrt{x-3}\) and write it in interval notation.

The domain is \([3,\infty)\). The endpoint \(3\) is included because \(\sqrt{0}\) is defined. Infinity is not included, so the interval ends with a parenthesis.

Example 5: Domain of a rational function

Find the domain of \(g(x)=\frac{1}{x-4}\) and write it in interval notation.

The domain is \((-\infty,4)\cup(4,\infty)\). The value \(4\) is excluded because it would make the denominator zero. The interval breaks at \(4\), so a union is required.

Open intervals, closed intervals, and half-open intervals

An open interval excludes both endpoints. The notation \((3,8)\) means \(3 < x < 8\). The values \(3\) and \(8\) are boundaries, but they are not members of the set. On a number line, an open interval is often shown with open circles at the endpoints and shading between them. Open intervals appear when inequalities use \(<\) or \(>\), when endpoints are not part of a domain, or when a graph approaches a value without reaching it.

A closed interval includes both endpoints. The notation \([3,8]\) means \(3 \le x \le 8\). On a number line, a closed interval is usually shown with filled circles at both endpoints. Closed intervals appear when a problem says “between 3 and 8 inclusive,” when a continuous function is restricted to a fixed segment, or when a solution includes equality at both boundaries.

A half-open interval includes one endpoint and excludes the other. The interval \([3,8)\) includes \(3\) but excludes \(8\). The interval \((3,8]\) excludes \(3\) but includes \(8\). These intervals are common in piecewise functions, class intervals, probability ranges, and domains where one boundary is allowed but another boundary is not. In tests and homework, half-open intervals are important because one wrong bracket can change the answer.

Using interval notation for domains and ranges

Interval notation is often used to write the domain and range of a function. The domain is the set of allowed input values, usually \(x\)-values. The range is the set of possible output values, usually \(y\)-values. If a graph extends forever left and right, its domain might be \((-\infty,\infty)\), which means all real numbers. If a graph starts at \(x=0\) with a filled dot and continues to the right forever, its domain is \([0,\infty)\).

For range, look vertically instead of horizontally. If a graph has a lowest point at \(y=-2\) and continues upward forever, the range is \([-2,\infty)\). If the lowest point is an open circle instead of a filled point, the range would be \((-2,\infty)\). The difference between a bracket and a parenthesis depends on whether the graph actually reaches that value.

For rational functions, interval notation is especially helpful because vertical asymptotes split the domain. For example, \(f(x)=\frac{1}{x+1}\) is undefined at \(x=-1\), so the domain is \((-\infty,-1)\cup(-1,\infty)\). The notation clearly shows that the function is defined on both sides of \(-1\), but not at \(-1\) itself.

Union and intersection in interval notation

The union of intervals includes values that are in at least one of the intervals. It is written with \( \cup \). For example, \([1,4]\cup[7,9]\) includes all values from \(1\) to \(4\), together with all values from \(7\) to \(9\). The gap between \(4\) and \(7\) is not included. Union notation is used when a solution set has multiple separate pieces.

The intersection of intervals includes only values that are in both intervals at the same time. It is written with \( \cap \). For example, \([1,6]\cap[4,10]=[4,6]\), because the overlap between the two intervals starts at \(4\) and ends at \(6\). Intersections appear when two conditions must both be true. In inequality language, intersection often corresponds to “and.”

Understanding union and intersection helps prevent a common mistake: combining intervals that should remain separate. For example, \((-\infty,2)\cup(5,\infty)\) cannot be simplified to \((-\infty,\infty)\), because the interval from \(2\) to \(5\) is missing. The union symbol does not automatically fill the gap. It only combines the values that are actually included in the listed intervals.

Common mistakes with interval notation

The first common mistake is using a bracket with infinity. Expressions such as \([0,\infty]\) and \([- \infty,3)\) are not correct because infinity is not a number that can be included. Always use a parenthesis beside \( \infty \) or \( -\infty \). The correct forms are \([0,\infty)\) and \((-\infty,3)\).

The second common mistake is reversing the order of endpoints. Interval notation should move from left to right on the number line. The smaller value goes first, and the larger value goes second. The interval \((7,2)\) is not a normal real-number interval because \(7\) is greater than \(2\). If the problem says \(2 < x < 7\), the interval is \((2,7)\), not \((7,2)\).

The third common mistake is confusing \(<\) with \(\le\). If the inequality includes equality, use a bracket. If it does not include equality, use a parenthesis. For example, \(x < 5\) becomes \((-\infty,5)\), while \(x \le 5\) becomes \((-\infty,5]\). That one bracket changes whether \(5\) is part of the solution set.

The fourth common mistake is forgetting the union symbol. If a solution set has two separated regions, both regions must appear in the final answer. For example, the solution \(x<-1\) or \(x>3\) is \((-\infty,-1)\cup(3,\infty)\). Writing only one of the intervals gives an incomplete answer. Writing \((-1,3)\) gives the opposite region.

The fifth common mistake is using interval notation before solving the inequality completely. For example, if the problem is \(2x+1<9\), you should first solve it: \(2x<8\), so \(x<4\). Only then convert it to \((-\infty,4)\). The calculator converts the final inequality or endpoint structure; it does not replace the algebra needed to isolate \(x\) in every possible equation.

How interval notation connects to calculus

In calculus, interval notation is used constantly. When finding where a function is increasing or decreasing, you often write answers such as \((-\infty,-1)\cup(3,\infty)\). When identifying concavity, you might write that a function is concave up on \((2,\infty)\) and concave down on \((-\infty,2)\). When writing the domain of a derivative, the interval notation may change depending on where the derivative exists.

Endpoints matter in calculus because many conclusions are stated over open intervals. For example, a function may be increasing on \((a,b)\), while endpoints are discussed separately. Critical points, discontinuities, and vertical asymptotes often split the number line into intervals. After testing each interval, students write the final result in interval notation. This is why a strong understanding of parentheses, brackets, and unions is essential before moving deeper into derivative applications.

If you are studying increasing and decreasing intervals, use this calculator together with a first-derivative sign chart. First find the critical values, then divide the number line into intervals, then determine the sign of \(f'(x)\) on each interval, and finally write the answer in interval notation. For a deeper calculus explanation, visit the Num8ers guide on determining intervals on which a function is increasing or decreasing.

Interval notation quick reference

Related Num8ers resources

Use interval notation with related algebra and calculus topics. These links are placed here because interval notation is commonly used when solving inequalities, writing domains and ranges, and describing function behavior.

Interval notation FAQs