Free Inequality to Interval Notation Calculator

The free online inequality to interval notation converter described here offers a straightforward way to translate between inequality statements and interval representations. It supports both directions: given an inequality you can obtain the interval (or union of intervals), and given an interval you can retrieve the equivalent inequality. Compound inequalities joined by "and" or "or" are also handled, making this interval notation converter a versatile tool for students, teachers, and anyone who needs to write an inequality in interval notation quickly and accurately.

Interval fundamentals

In mathematics, an interval is a subset of the real numbers that contains every value between two boundary numbers, called endpoints. The notation consists of the two endpoints separated by a comma, enclosed by either parentheses or square brackets. The numbers belonging to an interval are precisely those that satisfy the associated inequality (or inequalities). For example, the set defined by $2 < x < 5$ corresponds to the open interval $(2,5)$, while the interval $[-1, 4]$ is equivalent to the inequality $-1 \le x \le 4$.

Types of intervals

Intervals are classified by whether the endpoints are part of the set:

• Open interval $(a,b)$: both endpoints are excluded. This matches the inequality $a < x < b$.
• Closed interval $[a,b]$: both endpoints are included. This matches $a \le x \le b$.
• Half‑open intervals $(a,b]$ or $[a,b)$: exactly one endpoint is included. They correspond to $a < x \le b$ and $a \le x < b$ respectively.
• Unbounded intervals extend to infinity, such as $(a,\infty)$, $[a,\infty)$, $(-\infty,a)$, and $(-\infty,a]$. They represent one‑sided inequalities like $x > a$, $x \ge a$, $x < a$, and $x \le a$. Infinity always receives a parenthesis because it is not a number that can be included.

How the converter works

Using the tool is simple:

1. Select the conversion direction from the menu: Interval → Inequality or Inequality → Interval.
2. Depending on the mode:
   • In Interval mode, choose the interval type (open, closed, half‑open) and enter the two endpoints.
   • In Inequality mode, choose the inequality format (one‑sided, two‑sided, or compound) and input the inequality details.
3. Click "Calculate" or press Enter. The result appears immediately in the output field using standard notation.

The calculator automatically simplifies the result. For instance, if you enter the inequalities $x > 2$ and $x < 5$, it combines them into the two‑sided inequality $2 < x < 5$ and displays the interval $(2,5)$. This instant feedback makes it a practical tool for learning and verification.

Manual conversion: the essential mapping

The following table shows the direct correspondence between standard interval forms and the inequalities they represent. This mapping is the foundation for converting inequality to interval notation by hand.

The first four intervals are bounded (two finite endpoints); the last four are unbounded, extending to $\infty$ or $-\infty$. The key principle is: strict inequality signs ($<$ or $>$) correspond to parentheses; non‑strict signs ($\le$ or $\ge$) correspond to brackets.

To convert an inequality into interval notation manually:

1. Identify the variable (usually $x$) and the constant(s) on each side.
2. Write the lower bound (if any) and the upper bound (if any) from smallest to largest.
3. For a strict inequality use a parenthesis at that bound; for a non‑strict inequality use a bracket.
4. For one‑sided inequalities, use $-\infty$ (with a parenthesis) for the missing left bound or $\infty$ (with a parenthesis) for the missing right bound.
5. Separate the bounds with a comma and enclose them in the chosen brackets.

Example: Convert $x \le -2$.

• Left bound is absent → use $-\infty$ with a parenthesis.
• Right bound is $-2$ with a non‑strict inequality → use a bracket.
• Result: $(-\infty, -2]$.

Compound inequalities with "and"

When two inequalities are connected by and, a value must satisfy both to be part of the solution. The conversion depends on the direction and relative values of the bounds.

• Same direction both pointing right (e.g., $x > a$ and $x > b$): the more restrictive condition (larger endpoint) determines the interval. The bracket type follows the more restrictive inequality.
  • Example: $x \ge 2$ and $x > 3$ → $(3,\infty)$ because $x>3$ is more restrictive and strict.
  • Example: $x \ge 2$ and $x \ge 3$ → $[3,\infty)$ (both non‑strict, larger endpoint is 3).
• Same direction both pointing left (e.g., $x < a$ and $x < b$): the more restrictive condition (smaller endpoint) determines the interval. The bracket type follows the more restrictive inequality.
  • Example: $x < 4$ and $x \le 2$ → $(-\infty,2]$.
• Opposite directions (e.g., $x > a$ and $x < b$): the result is a bounded interval, provided $a < b$. The brackets are determined by the strictness of each inequality.

The following table summarizes the bounded case (assuming $a < b$):

• If $a = b$, the solution is a single point $\{a\}$ only when both inequalities are non‑strict; otherwise it is the empty set $\emptyset$.
• If $a > b$, the intersection is always empty ($\emptyset$).

Compound inequalities with "or"

With or, any value that satisfies at least one of the inequalities is included.

• Same direction both pointing right: the less restrictive condition (smaller endpoint) determines the interval. The bracket type follows the less restrictive inequality.
  • Example: $x > 1$ or $x > 4$ → $(1,\infty)$.
• Same direction both pointing left: the less restrictive condition (larger endpoint) determines the interval.
  • Example: $x \le 0$ or $x \le -2$ → $(-\infty,0]$.
• Opposite directions (e.g., $x \ge a$ or $x \le b$): the result is typically a union of two disjoint unbounded intervals, provided $a > b$. The following table covers the common cases (assuming $a > b$):

• If $a < b$ (the intervals overlap), the union covers all real numbers $\mathbb{R}$.
• If $a = b$, the union is $\mathbb{R}$ when at least one inequality is non‑strict, and $\mathbb{R} \setminus \{a\}$ (all real numbers except $a$) when both are strict.

Why use this converter?

The calculator eliminates the risk of misapplying bracket rules, especially with compound inequalities. It serves as a reliable interval notation converter that can handle bounded, unbounded, and compound cases, instantly delivering the correct result. Whether you are solving inequalities, graphing solution sets, or studying set notation, this tool provides a quick cross‑check and a clear illustration of how to express inequality in interval notation.

Manual conversions can always be verified with the calculator to ensure accuracy, making it an indispensable resource for mastering this fundamental mathematical notation.