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Interval Notation Practice Problems

Fifteen hand-built problems on translating between inequalities, graphs, and interval notation in every direction, with full solutions. Each shows the bracket-versus-parenthesis choice and the reasoning behind it.

  1. 1.

    Write in interval notation.

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  2. 2.

    Write in interval notation.

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  3. 3.

    Write in interval notation.

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  4. 4.

    Write "all real numbers" in interval notation.

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  5. 5.

    Convert to an inequality.

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  6. 6.

    Convert to an inequality.

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  7. 7.

    Write or in interval notation.

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  8. 8.

    A number line shows a filled dot at , shading to the right. Write the interval.

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  9. 9.

    Convert to an inequality and describe the graph.

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    ; open dots at and , shaded between.

  10. 10.

    Write and as a single interval.

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  11. 11.

    Write "" in interval notation.

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  12. 12.

    Write or in interval notation.

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  13. 13.

    Convert to a compound inequality.

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    or

  14. 14.

    A number line shows an open dot at , a filled dot at , shaded between. Write the interval.

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    , i.e. .

  15. 15.

    Write "all real numbers except those in " in interval notation.

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Written by
MPhil in Mathematics, Founder of Mathovia
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Interval Notation Practice Problems

The Interval Notation lesson is the full rulebook; this set is 15 conversion problems to practice on, each fully solved.

How to use this set

Convert each item — inequality, graph, or interval — into the other forms, matching brackets to the inclusion of each endpoint. Put your answer in the box and press Check Answers. Because this topic is about notation, the checker accepts every equivalent spelling — [0, 6) U (10, inf), [0, 6) ∪ (10, ∞), and the matching inequality all count.

Common mistakes to watch for

A bracket on infinity. Always a parenthesis.

Mismatched bracket and symbol. is .

Larger number first. is wrong.

Turning an “or” set into one interval instead of a union.

Where to go next

This is the answer format for every inequality lesson: Linear Inequalities, Compound Inequalities, and Polynomial Inequalities. For the full rulebook, see the Interval Notation lesson.

Frequently Asked Questions

When do I use a bracket versus a parenthesis?+

A bracket includes the endpoint (goes with , , a filled dot). A parenthesis excludes it (goes with , , an open dot). Infinity always takes a parenthesis.

Why does infinity always get a parenthesis?+

A bracket marks an included endpoint, and infinity is not a number — there is nothing there to include. It describes an unbounded direction.

What does the union symbol mean?+

joins two or more intervals when the solution set is not one connected piece, like .

How do I write 'all real numbers' and 'the empty set'?+

All real numbers is . The empty set is or .

Can an interval be written as [5, 2]?+

No — the smaller number always goes first. Rewrite as , which is .

Which notation does the checker treat as correct?+

All of the equivalent ones, since this topic is about translating between them. [0, 6) U (10, inf), [0, 6) ∪ (10, ∞) and the matching inequality are the same answer here. What does matter is the bracket: [ and ( are compared, because that is the distinction the exercise is teaching.

Calculators for this topic

Interval Notation Calculator

More practice problems

Linear Inequalities Practice Problems Compound Inequalities Practice Problems Polynomial Inequalities Practice Problems