SAT Math · Algebra

Interval notation

Q: What is interval notation?

Interval notation is a short way to write a set of numbers using its endpoints. Brackets mean an endpoint is included, parentheses mean it is not, so [−1, 4) is every number from −1 up to but not including 4.

Q: What is the difference between a bracket and a parenthesis?

A square bracket means the endpoint is part of the set, matching ≤ or ≥. A parenthesis means the endpoint is left out, matching . On a number line, brackets are filled dots and parentheses are open dots.

Q: Why does infinity always use a parenthesis?

Infinity is not a number, so a set can never include it as an endpoint. That is why x ≥ 3 is written [3, ∞), with a bracket on 3 and a parenthesis on infinity.

Q: How do you write “or” in interval notation?

Write each piece as its own interval and join them with the union symbol ∪. For example, x < −2 or x ≥ 3 becomes (−∞, −2) ∪ [3, ∞).

Q: Is interval notation on the SAT?

The SAT tests inequalities and solution sets in its Algebra domain, but it usually writes them as inequalities instead of interval notation. Knowing interval notation still helps, because the thinking about included and excluded endpoints is the same.

Q: How can Aristotle help with SAT algebra?

Aristotle is a voice AI SAT tutor. Your child talks through an inequality out loud, and Aristotle asks guiding questions instead of giving the answer, starting from the step where their reasoning broke. It tracks 113 SAT Math skills, including linear inequalities.

Brackets, parentheses, and infinity: how to write the solution to an inequality.

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Updated October 30, 2026

$[\;\text{included} \qquad (\;\text{not included}$

Filled dots are brackets, open dots are parentheses, arrows go to infinity.

What is interval notation?

-1 \le x < 4 \;\longrightarrow\; [-1,\ 4)

A shorthand for a stretch of the number line. Write the two endpoints, smaller first, and let the shape of each bracket say whether that endpoint belongs: a square bracket includes it, a round parenthesis leaves it out.

Filled dot at −1 (included), open dot at 4 (left out).

Square is a wall, round slides past

x \le 3 \to (-\infty,\ 3]

x > 3 \to (3,\ \infty)

A square corner is a wall: 3 is in. A round corner is a curve the shading slides past: 3 is out. It mirrors the number line exactly, a filled dot for $\le$ or $\ge$ and an open dot for $<$ or $>$ .

Same endpoint, 3; the bracket shape says whether 3 counts.

Infinity is never reached, and “or” means union

x < -2 \text{ or } x \ge 3

\longrightarrow\; (-\infty,\ -2) \cup [3,\ \infty)

There is no last number, so you can never include infinity; its side is always a parenthesis. When a solution comes in two separate pieces, write each piece as its own interval and join them with the union sign.

Two separate pieces, two intervals, joined by ∪.

How to read and write it

Answer each question in your head, then scroll for the answer.

Think it through

Question 1
$x \ge 3$ . Is 3 in the set? Bracket or parenthesis?
Answer
Yes, $\ge$ includes it, so a bracket: $[3, \ldots$
Question 2
What goes on the other end, and why?
Answer
$\infty$ with a parenthesis, since no number reaches infinity: $[3, \infty)$ .
Question 3
$x < -2$ or $x \ge 3$ . One interval or two?
Answer
Two pieces joined with $\cup$ : $(-\infty, -2) \cup [3, \infty)$ .

Try it yourself

Work it out first, then check your answer. SAT-style questions, not official College Board questions.

Question 1

Solve $3x - 5 < 7$ and write the solution in interval notation.

Question 2

Solve $-2x + 1 \ge 9$ and write the solution in interval notation.

Question 3

Solve $-3 < 2x + 1 \le 7$ and write the solution in interval notation.

Question 4

How many integers are in the interval $[-2, 3)$ ?

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The rules

Square bracket [ ]

$\le \text{ or } \ge \;\;\to\;\; [\;\;]$

The endpoint is part of the set. On a number line, it is a filled-in dot.

Parenthesis ( )

$< \text{ or } > \;\;\to\;\; (\;\;)$

The endpoint is not part of the set. On a number line, it is an open dot.

Infinity

$(-\infty, a] \quad\text{or}\quad [a, \infty)$

$\infty$ and $-\infty$ always get a parenthesis.

Union for “or”

$(-\infty, -2) \cup [3, \infty)$

Two separate pieces join with $\cup$ . This one means $x < -2$ or $x \ge 3$ .

Inequalities and their interval notation
Inequality	Interval notation	Endpoints
$x > a$	$(a, \infty)$	$a$ not included
$x \ge a$	$[a, \infty)$	$a$ included
$x < b$	$(-\infty, b)$	$b$ not included
$x \le b$	$(-\infty, b]$	$b$ included
$a < x \le b$	$(a, b]$	only $b$ included
$a \le x \le b$	$[a, b]$	both included
all real numbers	$(-\infty, \infty)$	no endpoints

Try it

Build an interval

Move each end and choose whether it is included.

Left end−1

Right end4

Inequality−1 ≤ x < 4

Interval notation[−1, 4)

Slide an end all the way out for infinity. It always gets a parenthesis.

Common mistakes

Putting a bracket next to infinity

$[3, \infty]$ is never right. Infinity always takes a parenthesis: $[3, \infty)$ .

Forgetting to flip the sign

Dividing $-2x \ge 8$ by $-2$ gives $x \le -4$ , not $x \ge -4$ .

Writing the larger number first

$(4, -1)$ is not an interval. The smaller endpoint always comes first: $(-1, 4)$ .

Using one interval for “or”

$x < -2$ or $x \ge 3$ is two separate pieces. Write $(-\infty, -2) \cup [3, \infty)$ , not $(-2, 3]$ .

How it shows up on the SAT

Algebra is about 35% of SAT Math. The SAT usually writes solution sets as inequalities, but the skill is the same: solve, flip when needed, and track each endpoint.

What Aristotle tracks in linear inequalities in one or two variables

5 SAT Math skills, including:

Writing and using linear inequalities
Writing linear inequalities to model constraints
Explaining parts and solutions of linear inequalities
Interpreting points as solutions to linear inequalities

What is interval notation?

Interval notation is a short way to write a set of numbers using its endpoints. Brackets mean an endpoint is included, parentheses mean it is not, so [−1, 4) is every number from −1 up to but not including 4.

What is the difference between a bracket and a parenthesis?

Why does infinity always use a parenthesis?

How do you write “or” in interval notation?

Is interval notation on the SAT?

How can Aristotle help with SAT algebra?

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