Math July 13, 2026 · 8 Min Read

Inequality to Interval Notation Calculator – Guide & Formulas

Convert any inequality to interval notation instantly. Enter compound inequalities, absolute value inequalities, or simple inequalities and get proper interval notation.

Convert any inequality to interval notation instantly with our free calculator. Enter your inequality and get proper bracket and parenthesis notation with number line display.

Key Takeaway

Use the free Inequality to Interval Notation Calculator to convert any inequality to interval notation instantly. enter compound inequalities, absolute value inequalities, or simple inequalities and get proper interval notation. Get instant results with step-by-step explanations.

How to Use the Inequality to Interval Notation Calculator

  1. Enter your inequality using standard math notation (e.g., x > 3, -2 <= x < 5, |x| > 4).
  2. Use <= for ≤ and >= for ≥ symbols.
  3. Review the converted interval notation.
  4. Check the number line visualization for clarity.

The Formula

Strict inequalities (< or >) use parentheses: (a, b). Non-strict inequalities (≤ or ≥) use brackets: [a, b). Mixed: a ≤ x < b becomes [a, b).

Variable Definitions

  • ( ): Parentheses indicate the endpoint is NOT included (strict inequality: < or >)
  • [ ]: Brackets indicate the endpoint IS included (non-strict inequality: ≤ or ≥)
  • ( ) or [ ): Mixed notation for one strict and one non-strict endpoint
  • ∪: Union symbol: combines multiple intervals into one solution set
  • ∞: Infinity: always uses a parenthesis, never a bracket

Converting -2 ≤ x < 5 to Interval Notation

Convert a compound inequality to interval notation.

  1. Step 1: Identify the lower bound: x ≥ -2 means -2 is included → use bracket [.
  2. Step 2: Identify the upper bound: x < 5 means 5 is not included → use parenthesis ).
  3. Step 3: Combine: the interval notation is [-2, 5).
  4. Step 4: On a number line: closed circle at -2, open circle at 5, shaded between.

Frequently Asked Questions

What is interval notation?

Interval notation is a mathematical way to describe a range of numbers. It uses parentheses ( ) for excluded endpoints and brackets [ ] for included endpoints. For example, (2, 5) means all numbers between 2 and 5, not including either.

When do I use parentheses vs brackets?

Use parentheses ( ) for strict inequalities: x > 2 or x < 5. Use brackets [ ] for non-strict inequalities: x ≥ 2 or x ≤ 5. Infinity and negative infinity always get parentheses.

What does the union symbol mean?

The union symbol ∪ combines two separate intervals. For example, x < -2 or x > 3 is written as (-∞, -2) ∪ (3, ∞). It means the solution includes numbers in either interval.

How do I convert absolute value inequalities?

For |x| > a, the solution is (-∞, -a) ∪ (a, ∞). For |x| < a, the solution is (-a, a). Handle compound absolute inequalities by splitting into two separate inequalities.

What about infinity?

Infinity (∞) is not a real number, so it always gets a parenthesis: (a, ∞) or (-∞, b). You can never use a bracket with infinity.

Can interval notation include just one number?

Yes, if the inequality is x = 3, the interval notation is [3, 3], which contains exactly one point. This is called a degenerate or singleton interval.

What is the difference between interval notation and set-builder notation?

Interval notation: (2, 5). Set-builder notation: {x | 2 < x < 5}. They describe the same set but interval notation is more compact.

How do I handle no solution or all real numbers?

No solution is written as ∅ (empty set). All real numbers is (-∞, ∞).