Inequality Interval Notation Calculator

What interval notation is really showing

Interval notation is a compact way to write all numbers that satisfy an inequality. Instead of writing x > 3 in words, interval notation writes (3, ∞). The left parenthesis means 3 is not included, and infinity always uses a parenthesis because infinity is not a number you can actually reach.

OpenStax’s Algebra 1 material explains that inequalities can be represented using interval notation and shows the basic example x > 3 as (3,∞). That is the same endpoint logic this calculator uses for single, bounded, and outside intervals.

Parentheses vs brackets

The most important part of interval notation is knowing whether the endpoint is included. A strict inequality like < or > uses parentheses because the endpoint is not part of the answer. An inclusive inequality like ≤ or ≥ uses a square bracket because the endpoint is included.

Worked example: solve and write interval notation

The bracket after 5 is important. The original inequality uses ≤, so x can equal 5. If the problem had been 3x - 5 < 10, the answer would be (-∞, 5) instead.

Why the inequality sign flips when dividing by a negative

When you divide or multiply both sides of an inequality by a negative number, the direction of the inequality reverses. For example, -2x < 8 becomes x > -4, not x < -4. Forgetting this sign flip is one of the most common reasons students get the correct number but the wrong interval.

Common questions

• What is interval notation?
  Interval notation is a way to describe a set of numbers using endpoints, parentheses, brackets, infinity symbols, and sometimes the union symbol. For example, x > 4 becomes (4, ∞). This means every number greater than 4, but not 4 itself. It is commonly used in algebra because it is shorter and cleaner than writing long inequality statements.
• When do I use parentheses in interval notation?
  Use parentheses when an endpoint is not included in the solution. Strict inequalities such as x < 7 and x > 7 use parentheses because x cannot equal 7. Parentheses are also always used next to infinity and negative infinity because infinity is not a fixed number or endpoint.
• When do I use square brackets in interval notation?
  Use square brackets when the endpoint is included in the answer. Inequalities with ≤ or ≥ include the boundary value, so x ≤ 5 becomes (-∞, 5] and x ≥ 5 becomes [5, ∞). A bracket tells the reader that the endpoint itself is part of the solution set.
• What does the union symbol mean in interval notation?
  The union symbol ∪ means the solution has separate pieces that are both allowed. For example, x < -2 or x > 3 becomes (-∞, -2) ∪ (3, ∞). This is common with outside inequalities and absolute value inequalities with greater-than signs.
• Why does infinity always use parentheses?
  Infinity is not a number that can be included. It only shows that the interval continues forever in one direction. Because there is no final endpoint called infinity, interval notation always uses parentheses next to ∞ or -∞, never brackets.
• How do I convert a double inequality into interval notation?
  A double inequality like 2 < x ≤ 9 means the solution is between 2 and 9. The left endpoint is open because 2 is not included, while the right endpoint is closed because 9 is included. The interval notation is (2, 9].
• What is the interval notation for all real numbers?
  All real numbers are written as (-∞, ∞). Both sides use parentheses because infinity and negative infinity are not real endpoints. This answer usually appears when an inequality is true for every possible value of x.
• What is the interval notation for no solution?
  No solution is often written as ∅, which means the empty set. It appears when no value of x can make the inequality true, such as x < 2 and x > 5 at the same time.