Understand the idea.
Endpoints carry information
An interval describes a continuous set of real numbers. A square bracket includes a finite endpoint; a round bracket excludes it. Translate each endpoint separately: means , including but excluding .
Infinity records an unbounded direction, not a real number in the set, so it always takes a round bracket. Intervals contain real values between their endpoints; restrict to integers separately if the question requires an integer domain.
Combine sets, not just symbols
An intersection requires membership in both sets, so retain only their overlap. A union requires membership in at least one set, so combine all the allowed regions, including any separated pieces.
Test a boundary value against the original conditions before choosing its bracket. For , the allowed values lie below or above ; neither endpoint belongs, and the middle region must remain excluded.
See the method in action.
Two simultaneous bounds
Write in interval notation and state whether each endpoint is included.
Worked solution
- 1
Interpret the lower endpoint.
The strict inequality excludes , so the interval starts with a round bracket.
- 2
Interpret the upper endpoint.
Equality is allowed at , so the interval ends with a square bracket.
- 3
Combine the simultaneous bounds.
The interval includes every real value between the bounds, with only the upper endpoint included.
; -1 excluded and 4 included.
An intersection
Find .
Worked solution
- 1
Write the two membership conditions.
An intersection requires both inequalities to hold.
- 2
Keep the stricter lower and upper bounds.
The second interval excludes ; the first interval excludes .
- 3
Express the overlap as an interval.
Both limiting endpoints are excluded from the common region.
.
Try it yourself.
Check your understanding
Write or in interval notation.
Show answer & reasoning
.
The word “or” gives a union: . Zero is excluded by , while is included by .
Keep practising
6 worksheet questions · 14 marks
Read the questions online
- Write as an interval.2 marks
- Write as an inequality.2 marks
- Find .2 marks
- Find .2 marks
- Solve for real and write intervals.3 marks
- Describe 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 as an interval.2 marks
Include 2 and exclude 7.
. - Write as an inequality.2 marks
The finite endpoint is included and the interval extends left without a bound.
. - Find .2 marks
The common numbers run from 1 to 4, and both endpoints lie in both sets.
. - Find .2 marks
The intervals overlap, so the union has no gap. Both outside endpoints are included.
. - Solve for real and write intervals.3 marks
The square exceeds 9 when or ; equality is excluded.
. - Describe 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.
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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 ↗