This interval notation calculator, provided by Hesapstan, converts interval notation to inequality form and turns a one-variable linear inequality into interval notation. It handles open and closed endpoints, infinity, unions of up to three intervals, the empty set and all-real-number cases.
What does this calculator do?
The interval notation calculator works in two directions: it reads an interval and writes the corresponding inequality, or it solves a linear inequality and writes the solution in interval notation.
- Interval to inequality: for example, (2, 7] becomes 2 < x ≤ 7.
- Inequality to interval: for example, 3x + 2 ≤ 11 becomes x ≤ 3, then (-∞, 3].
- Union support: up to three interval pieces can be entered.
- Special cases: empty set and all real numbers are shown with explicit labels.
This page is the general bidirectional interval notation tool. A separate landing page opens directly on the inequality-to-interval direction for users who searched for that specific task, but the calculation logic is shared.
What is interval notation?
Interval notation is a compact way to describe a set of real numbers on the number line using brackets, parentheses and endpoints.
Instead of writing a full inequality such as 1 ≤ x < 5, interval notation writes the same solution as [1, 5). The left bracket means 1 is included; the right parenthesis means 5 is not included.
This notation is common in algebra, functions, domains and ranges, solution sets and number-line questions.
What do brackets and parentheses mean?
In interval notation, the bracket type tells you whether an endpoint belongs to the solution set.
- [a, b] is a closed interval; both a and b are included.
- (a, b) is an open interval; neither endpoint is included.
- [a, b) includes a but excludes b.
- (a, b] excludes a but includes b.
A square bracket corresponds to ≤ or ≥. A parenthesis corresponds to < or >. The most common mistake is using a parenthesis when the endpoint should actually be included.
How is infinity written in interval notation?
Infinity is never included as an endpoint, so -∞ and ∞ always use parentheses in interval notation.
- x < 4 is written as (-∞, 4).
- x ≤ 4 is written as (-∞, 4].
- x > -2 is written as (-2, ∞).
- x ≥ -2 is written as [-2, ∞).
Do not write [∞] or [-∞]. Infinity is a direction, not a number that can be included in the set.
How interval to inequality conversion works
Interval to inequality conversion reads each endpoint and turns the bracket type into the correct inequality sign.
- Enter the lower and upper endpoints.
- Choose open or closed endpoint symbols.
- Use the infinity option for unbounded intervals.
- Add additional intervals if the solution is a union.
- Read the resulting inequality or joined set of inequalities.
For example, [-3, 2) becomes -3 ≤ x < 2. The interval (-∞, 5] becomes x ≤ 5.
How inequality to interval conversion works
Inequality to interval conversion isolates x in a linear inequality and then writes the solution side as an interval.
This calculator uses a one-row linear form: ax + b [operator] c. The operator may be <, ≤, > or ≥. Once x is isolated, the solution becomes an interval with infinity on one side.
When both sides are divided by a negative coefficient, the inequality direction reverses. The calculator applies this rule automatically.
Example: convert an interval to an inequality
The interval [1, 6) includes the left endpoint and excludes the right endpoint, so it becomes 1 ≤ x < 6.
- Left endpoint 1 with a square bracket: x ≥ 1.
- Right endpoint 6 with a parenthesis: x < 6.
- Combined inequality: 1 ≤ x < 6.
For a union example, (-∞, -2) ∪ [3, 5] reads as x < -2 or 3 ≤ x ≤ 5. The union symbol means the solution set has separate pieces.
Example: convert a linear inequality to interval notation
The inequality 3x + 2 ≤ 11 is solved first, then the result is written as interval notation.
- 3x + 2 ≤ 11
- 3x ≤ 9
- x ≤ 3
- Interval notation: (-∞, 3]
If the inequality is -2x + 4 > 10, then -2x > 6 and x < -3. Because the calculation divides by a negative number, the inequality sign flips. The interval is (-∞, -3).
How empty set and all real numbers are shown
Some linear inequalities have no solution, while others are true for every real number. The calculator shows these special cases explicitly instead of forcing them into a misleading ordinary interval.
- A contradiction gives the empty set: ∅.
- An always-true inequality gives all real numbers.
- All real numbers may also be written as (-∞, ∞).
For example, 0x + 5 < 3 has no solution, so the result is ∅. The inequality 0x + 2 ≤ 2 is true for every real x, so the result is all real numbers.
What are the calculator limits?
This calculator is built for interval notation and one-variable linear inequality conversion. It is not a general algebra solver or graphing engine.
- Interval to inequality mode supports a union of up to three intervals.
- Inequality to interval mode targets linear inequalities of the form ax + b [operator] c.
- It does not solve quadratic, absolute-value, rational or two-variable inequalities on this page.
- It focuses on notation conversion, not detailed number-line graphing.
- Invalid intervals, such as a lower endpoint greater than the upper endpoint, are rejected.
A quadratic or more complex inequality cannot be safely treated as a simple linear inequality. If the expression is not linear, the interval result may require a different sign analysis method.
Frequently Asked Questions
How do I write interval notation?
Write the endpoints in order and use brackets or parentheses to show whether each endpoint is included. Included endpoints use square brackets; excluded endpoints use parentheses.
What is the difference between open and closed intervals?
An open interval excludes its endpoint. A closed interval includes it. For example, (2, 5) excludes 2 and 5, while [2, 5] includes both.
Why does infinity always use parentheses?
Infinity is not an actual number or endpoint, so it cannot be included. That is why -∞ and ∞ always use parentheses.
Can this calculator handle unions of intervals?
Yes. In interval-to-inequality mode, it supports unions of up to three interval pieces.
Does this solve quadratic inequalities?
No. The inequality-to-interval mode is for linear inequalities in one variable. Quadratic inequalities need sign analysis around roots.