Interval Notation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Interval Notation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A shorthand for writing all real numbers in a range, using parentheses for excluded endpoints and square brackets for included endpoints.

Parentheses mean the endpoint is NOT included; square brackets mean it IS included. For example, $(2, 5]$ means $2 < x \le 5$ .

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Interval notation names a range of real numbers with square brackets at endpoints that are included and parentheses at endpoints that are excluded.

Common stuck point: The procedure for interval notation is the easy part; the trap is using a bracket on infinity. Asking "Am I describing a continuous range of reals where I must mark each endpoint as included (bracket) or excluded (parenthesis)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I describing a continuous range of reals where I must mark each endpoint as included (bracket) or excluded (parenthesis)?

Worked Examples

Example 1

easy

Write

x > 3

in interval notation.

Open circle at 3 (not included); arrow extends right

Answer

(3, \infty)

First step

x > 3

means all real numbers greater than 3.

Full solution

2
3 is not included (strict inequality), so use a parenthesis: $(3$ .
3
There is no upper bound, so use $\infty)$ .
4
Interval: $(3, \infty)$ .

In interval notation, parentheses

(\,)

mean the endpoint is NOT included, and brackets

[\,]

mean it IS included. Infinity always gets a parenthesis.

Example 2

medium

Write

-2 \leq x < 5

in interval notation.

Filled dot at −2 (included, ≤); open circle at 5 (excluded, <)

Example 3

medium

Write the domain of

g(x) = \dfrac{1}{x-5}

in interval notation.

Example 4

hard

Solve

\dfrac{x+1}{x-2} < 0

and write the solution in interval notation.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Write

x \leq 7

in interval notation.

Filled dot at 7 (included, ≤); arrow extends left

Example 2

medium

Convert the interval

[0, 4)

to inequality notation.

Read off the inequality: [0, 4) → bracket at 0 means ≥, parenthesis at 4 means <

Example 3

easy

Write

x>3

in interval notation.

Open circle = endpoint excluded; ray to the right = no upper bound

Example 4

easy

Write

x\le 5

in interval notation.

Filled dot at 5 (≤ includes the endpoint)

Example 5

easy

Write

2<x<7

in interval notation.

Both endpoints open (excluded); all values strictly between 2 and 7

Example 6

easy

Write

1\le x\le 4

in interval notation.

Filled dots at both ends; the closed interval includes every point from 1 to 4

Example 7

easy

Write

-3\le x<2

in interval notation.

Filled at −3 (≤ includes it); open at 2 (< excludes it)

Example 8

easy

Convert the interval

[0,\infty)

back to an inequality.

Example 9

easy

Write 'all real numbers' in interval notation.

Example 10

easy

Convert

(−2,5]

to an inequality.

Open at −2 (excluded, <); filled at 5 (included, ≤)

Example 11

medium

Write the solution of

2x-1\ge5

in interval notation.

Solve 2x − 1 ≥ 5 → x ≥ 3; filled dot because equality is included

Example 12

medium

Write the union of

x<0

x>2

in interval notation.

Example 13

medium

Express the intersection of

[1,5]

and

[3,8]

in interval notation.

[1, 5] ∩ [3, 8] = [3, 5]: the overlap runs from the larger left end to the smaller right end

Example 14

medium

Write the domain of

f(x)=\sqrt{x-4}

in interval notation.

Example 15

medium

Write the solution of

-3<2x+1\le7

in interval notation.

Solution of −3 < 2x+1 ≤ 7 simplified to −2 < x ≤ 3

Example 16

medium

Convert

(-\infty,-1]\cup[4,\infty)

to an inequality statement.

Example 17

medium

Write the set of

x

with

0\le x\le10

excluding

x=5

in interval notation.

Example 18

medium

A temperature must stay at or above

-5^\circ

and below

30^\circ

. Write the interval.

'At or above −5°' → filled dot; 'below 30°' → open circle

Example 19

medium

Write the solution of

5-2x\ge1

in interval notation.

Example 20

challenge

Solve

\frac{x-2}{x+1}\ge0

and write the solution in interval notation.

Example 21

challenge

Write the solution set of

x^2\le9

in interval notation.

x² ≤ 9 means −3 ≤ x ≤ 3; both endpoints included because equality holds at x = ±3

Example 22

challenge

Write the solution of

|x-1|>3

in interval notation.

Example 23

easy

Write

x \ge -2

in interval notation.

Filled dot at −2 (≥ includes it); bracket in interval notation becomes a filled circle on the number line

Example 24

easy

Write

-4 < x < 4

in interval notation.

Both endpoints open; strict < on each side means neither −4 nor 4 is included

Example 25

easy

Write

x < 0

in interval notation.

Example 26

easy

Convert

(-3, 7]

to inequality form.

( at −3 → strictly greater than; ] at 7 → less than or equal to

Example 27

easy

Write the empty set as an interval expression.

Example 28

medium

Solve

3x - 5 \le 7

and write the solution in interval notation.

Example 29

medium

Solve

-2 < x + 1 < 5

and write the solution in interval notation.

Subtract 1 throughout: −3 < x < 4; both endpoints excluded

Example 30

medium

Write the union

x \le -1

x > 2

in interval notation.

Example 31

medium

Write the domain of

f(x)=\sqrt{2x-6}

in interval notation.

Example 32

medium

Find the intersection

[2, 7) \cap (5, 10]

as an interval.

Example 33

medium

Solve

5 - 2x > 1

and write the solution in interval notation.

Example 34

medium

Express 'numbers from

-1

6

, excluding

3

' in interval notation.

Example 35

medium

Write the union

[1, 4] \cup [3, 7]

as a single interval.

Example 36

hard

Solve

x^2 - 4x \le 0

and write the solution in interval notation.

x(x−4) ≤ 0 holds between the roots 0 and 4; both roots included (equality satisfied)

Example 37

hard

Solve

|2x - 3| < 5

and write the solution in interval notation.

Example 38

hard

Solve

|x + 2| \ge 3

and write the solution in interval notation.

Example 39

hard

Solve

x^2 > 16

and write the solution in interval notation.

Example 40

hard

Write the domain of

h(x) = \sqrt{x^2 - 9}

in interval notation.

Example 41

challenge

Solve

\dfrac{x^2 - 4}{x + 1} \ge 0

and write the solution in interval notation.

Example 42

challenge

Write the domain of

f(x) = \ln(4 - x^2)

in interval notation.

Example 43

challenge

Solve

|x - 1| + |x + 2| \le 5

and write the solution in interval notation.

← Back to Interval Notation Practice Mode

Related Concepts

Inequalities Solution Set Number Line

Background Knowledge

These ideas may be useful before you work through the harder examples.

inequalitiessolution setnumber line