Math Practice

Interval Notation Practice Problems

Q: When do I use a bracket versus a parenthesis?

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

Q: 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.

Q: What does the union symbol mean?

\(\cup\) joins two or more intervals when the solution set is not one connected piece, like \((-\infty, -2) \cup (3, \infty)\).

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

All real numbers is \((-\infty, \infty)\). The empty set is \(\varnothing\) or \(\{\ \}\).

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

No — the smaller number always goes first. Rewrite \(5 \ge x > 2\) as \(2 < x \le 5\), which is \((2, 5]\).

Q: Why is this a curated set?

The answers are notation, not single numbers, so an automatic checker can't verify them. Each problem has a full solution instead.

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.
Write \(x > 5\) in interval notation.
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\((5, \infty)\)
2.
Write \(-3 \le x \le 2\) in interval notation.
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\([-3, 2]\)
3.
Write \(x \le -4\) in interval notation.
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\((-\infty, -4]\)
4.
Write "all real numbers" in interval notation.
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\((-\infty, \infty)\)
5.
Convert \((-\infty, 3)\) to an inequality.
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\(x < 3\)
6.
Convert \([2, 7)\) to an inequality.
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\(2 \le x < 7\)
7.
Write \(x < -1\) or \(x \ge 4\) in interval notation.
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\((-\infty, -1) \cup [4, \infty)\)
8.
A number line shows a filled dot at \(-2\), shading to the right. Write the interval.
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\([-2, \infty)\)
9.
Convert \((-4, 4)\) to an inequality and describe the graph.
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\(-4 < x < 4\); open dots at \(-4\) and \(4\), shaded between.
10.
Write \(x \ge -2\) and \(x < 7\) as a single interval.
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\([-2, 7)\)
11.
Write "\(x \neq 3\)" in interval notation.
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\((-\infty, 3) \cup (3, \infty)\)
12.
Write \(0 \le x < 6\) or \(x > 10\) in interval notation.
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\([0, 6) \cup (10, \infty)\)
13.
Convert \((-\infty, -1] \cup (2, \infty)\) to a compound inequality.
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\(x \le -1\) or \(x > 2\)
14.
A number line shows an open dot at \(1\), a filled dot at \(5\), shaded between. Write the interval.
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\((1, 5]\), i.e. \(1 < x \le 5\).
15.
Write "all real numbers except those in \([1, 4]\)" in interval notation.
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\((-\infty, 1) \cup (4, \infty)\)

Written by

Mathovia Team

Editorial Team

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

These problems are hand-written, not auto-generated. Convert each item — inequality, graph, or interval — into the other forms, matching brackets to the inclusion of each endpoint. Then compare with the solution.

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 .

Why is this a curated set?+

The answers are notation, not single numbers, so an automatic checker can't verify them. Each problem has a full solution instead.

[ )Interval Notation Practice Problems

Interval Notation Practice Problems

How to use this set

Common mistakes to watch for

Where to go next

Frequently Asked Questions

More practice problems

Interval Notation Practice Problems