Practice Interval Notation in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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 $2 < x \le 5$ .

Showing a random 20 of 50 problems.

Example 1

challenge

Solve

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

and write the solution in interval notation.

Example 2

medium

Write the domain of

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

in interval notation.

Example 3

challenge

Solve

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

and write the solution in interval notation.

Example 4

medium

Convert the interval

[0, 4)

to inequality notation.

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

Example 5

medium

Write the domain of

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

in interval notation.

Example 6

medium

Write the set of

x

with

0\le x\le10

excluding

x=5

in interval notation.

Example 7

hard

Solve

|2x - 3| < 5

and write the solution in interval notation.

Example 8

easy

Convert the interval

[0,\infty)

back to an inequality.

Example 9

medium

Solve

3x - 5 \le 7

and write the solution in interval notation.

Example 10

medium

Write the union

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

as a single interval.

Example 11

medium

Convert

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

to an inequality statement.

Example 12

easy

Write

2<x<7

in interval notation.

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

Example 13

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 14

medium

Solve

5 - 2x > 1

and write the solution in interval notation.

Example 15

easy

Convert

(−2,5]

to an inequality.

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

Example 16

medium

Write the solution of

5-2x\ge1

in interval notation.

Example 17

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 18

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 19

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 20

medium

Express 'numbers from

-1

6

, excluding

3

' in interval notation.

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Related Concepts

Inequalities Solution Set Number Line