Interval notation | Year 11

01 / LEARN

The idea, step by step

Endpoints carry information

An interval describes all real numbers between endpoints. A square bracket includes a finite endpoint; a round bracket excludes it. Thus $[2, 5)$ contains 2 but not 5. Infinity is not a real endpoint that can be included, so infinity always uses a round bracket.

Translate the inequality before drawing a number line: $x \geq a$ includes $a$ , while $x > a$ excludes it. Do not write an interval for the integers unless you explicitly intersect with $ℤ$ .

Combine sets, not just symbols

The union $A \cup B$ contains points in either set. The intersection $A \cap B$ contains points satisfying both conditions. For an “and” condition, look for the overlap; for an “or” condition, combine the permitted regions.

Some solution sets need two disjoint intervals. Write $(- \infty, - 2) \cup (2, \infty)$ for $∣ x ∣ > 2$ . The gap matters: it excludes every value from -2 to 2, including both endpoints.

A common trap

Using a square bracket next to infinity.

Instead: Infinity is unbounded direction, not an included real value.

02 / WORKED EXAMPLES

See the thinking

Example 1

Two simultaneous bounds

Write $- 1 < x \leq 4$ in interval notation and state whether each endpoint is included.

The inequality is strict at -1, so use a round bracket.
It includes 4, so use a square bracket.

(- 1, 4]

; -1 excluded and 4 included.

Example 2

An intersection

Find $[1, 6) \cap (3, 8]$ .

Both conditions require $x > 3$ because the second set excludes 3.
The first set requires $x < 6$ . Both restrictions must hold together.

(3, 6)

Check your understanding

Write $x < 0$ or $x \geq 2$ in interval notation.

Show answer

$(- \infty, 0) \cup [2, \infty)$ .

“Or” gives a union; only the finite endpoint 2 is included.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Write $2 \leq x < 7$ as an interval.
2 marks
Write $(- \infty, - 3]$ as an inequality.
2 marks
Find $[- 2, 4] \cap [1, 8)$ .
2 marks
Find $[0, 3) \cup [2, 5]$ .
2 marks
Solve $x^{2} > 9$ for real $x$ and write intervals.
3 marks
Describe $[1, 5] \cap ℤ$ and explain why it is not the full real interval.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Write $2 \leq x < 7$ as an interval.
2 marks
Include 2 and exclude 7.
$[2, 7)$ .
Write $(- \infty, - 3]$ as an inequality.
2 marks
The finite endpoint is included and the interval extends left without a bound.
$x \leq - 3$ .
Find $[- 2, 4] \cap [1, 8)$ .
2 marks
The common numbers run from 1 to 4, and both endpoints lie in both sets.
$[1, 4]$ .
Find $[0, 3) \cup [2, 5]$ .
2 marks
The intervals overlap, so the union has no gap. Both outside endpoints are included.
$[0, 5]$ .
Solve $x^{2} > 9$ for real $x$ and write intervals.
3 marks
The square exceeds 9 when $x < - 3$ or $x > 3$ ; equality is excluded.
$(- \infty, - 3) \cup (3, \infty)$ .
Describe $[1, 5] \cap ℤ$ and explain why it is not the full real interval.
3 marks
The intersection restricts the interval to integer points, excluding values such as 1.5.
${1, 2, 3, 4, 5}$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: geometric reasoning and real-number/proof material, regrouped into a school chapter; all exercises are original.

Official curriculum source ↗

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