Carry the language of sets onto the number line — continuous ranges become intervals and neighborhoods, the exact tool for a function's domain.
The sets so far had countable, listable members — the vowels, {2, 4, 6, 8}, a handful of integers. But the most important sets in all of analysis are continuous ranges: every real number between 1 and 5, with no gaps. You cannot list them — between any two you name there are infinitely many more — yet you can describe them perfectly: { x | 1 ≤ x ≤ 5 }, and far more compactly as the interval [1, 5]. The brackets carry one crucial bit of information — whether each endpoint is included — and with just a few symbols ([ ] ( ) ∞) we capture closed ranges, open ranges, half-open ranges, rays that run forever, and the neighborhood of a point. This is the precise language a function uses to state its domain, so this last lesson of Stage 19 opens the door straight into Stage 20.
Look at the inequality 1 ≤ x ≤ 5. Its solutions form a set — write it in set-builder form as { x | 1 ≤ x ≤ 5 }. But this set is unlike {1, 2, 3, 4, 5} in a deep way: it holds every real number in between — 1.5, π, 4.999…, √2 — infinitely many, packed together with no gaps. There is no first number after 1, no way to list them one by one.
So we change pictures. Instead of dots in a bubble we shade a band on the number line, and we give the band a compact name: an interval. You solved inequalities exactly like this back in Stage 12 (Solving linear inequalities); now the solution set itself gets a name and a notation.
A continuous range of real numbers is a set with infinitely many members and no gaps. We cannot roster it, so we shade it on the number line and write it as an interval.
The brackets are not decoration — they say whether each endpoint belongs to the set. A closed interval [a, b] = { x | a ≤ x ≤ b } includes both ends; we draw solid dots there. An open interval (a, b) = { x | a < x < b } excludes both ends; we draw hollow dots. So 1 ∈ [1, 5] — the closed end owns its boundary — but 1 ∉ (1, 5) — the open end stops just short.
The length of [a, b] and (a, b) is the same number, b − a. What changes between them is only the two boundary points — present or absent. Removing two points does not shorten a continuous range.
| Interval | Set-builder | Left end | Right end |
|---|---|---|---|
| [a, b] | { x | a ≤ x ≤ b } | solid (in) | solid (in) |
| (a, b) | { x | a < x < b } | hollow (out) | hollow (out) |
| [a, b) | { x | a ≤ x < b } | solid (in) | hollow (out) |
| (a, b] | { x | a < x ≤ b } | hollow (out) | solid (in) |
Is 5 ∈ (2, 5)? No — the right end is open, so 5 is excluded: 5 ∉ (2, 5). But 5 ∈ (2, 5] ✓, because that bracket is closed. The bracket is the whole question.
Nothing forces both ends to match. Mix them and you get a half-open interval: (a, b] = { x | a < x ≤ b } (open left, closed right) and [a, b) = { x | a ≤ x < b } (closed left, open right). Half-open intervals are everywhere — a bus that runs "from 9 up to but not including noon" is the interval [9, 12).
What about a range with no far end — "all numbers greater than 2"? Reach for the infinity symbol ∞: [a, ∞) = { x | x ≥ a }, (−∞, b) = { x | x < b }, and the whole line is ℝ = (−∞, ∞). On the picture an infinite end becomes an arrowhead running off the edge.
∞ is a direction, not a number. You can never reach it, so there is no point there to include — an infinite end is always open. Write [2, ∞), never [2, ∞]. The bracket beside ∞ is always round.
Now zoom in. Pick a single point a on the line and gather every number within a small distance δ ("delta") of it. That open interval is the neighborhood of a:
(a − δ, a + δ) = { x | |x − a| < δ }
It is just an open interval of radius δ centred at a — width 2δ, reaching δ to each side. The condition |x − a| < δ is exactly the absolute-value distance you met in Stage 5 (Opposites and absolute value): "x is less than δ away from a."
The neighborhood of radius 0.1 about 3 is (2.9, 3.1) = { x | |x − 3| < 0.1 } — every number that rounds the same way as 3 to one decimal, give or take.
Drop the centre itself and you get the punctured neighborhood { x | 0 < |x − a| < δ } — close to a but never equal to a. That picture is the secret engine behind "x approaches a," the idea limits and calculus are built on.
Intersection and union do not disappear when we move to the line — they turn into overlap and join. To find I₁ ∩ I₂, keep only the part common to both stretches; to find I₁ ∪ I₂, sweep up everything either one covers.
The one subtlety lives at a shared endpoint. Take [1, 5] ∩ (3, 8). The overlap runs from 3 to 5. At 3 the left interval is closed but the right is open, so to be in both a point must obey the stricter rule — 3 is excluded. At 5, that endpoint is the closed end of [1, 5] and lies strictly inside (3, 8) (since 3 < 5 < 8), so 5 is in both and stays included. Hence the answer is (3, 5]: at a tie the stricter (open) bracket wins for ∩.
For union the looser bracket wins. [1, 3) ∪ [3, 6]: the first stops just before 3, the second starts exactly at 3 and 3 ∈ [3, 6], so the gap is filled — the join is the single interval [1, 6]. And if the pieces never touch, the union stays in two parts and the intersection is empty: [2, 5] ∩ [7, 9] = ∅. This is precisely how we will combine the separate conditions that define a function's domain.
To intersect or unite intervals, compute the endpoints — don't eyeball. ∩: take the larger lower end and the smaller upper end; at a tie the stricter (open) bracket wins; empty if the result has nothing left. ∪ (when they overlap or touch): take the smaller lower and larger upper end; at a tie the looser (closed) bracket wins.
The interval (1, 5) and the coordinate point (1, 5) are written the same way but mean different things. Here the parentheses make an open interval — a set of real numbers on one line. In Stage 20 the same parentheses will name a point in the plane. Context tells them apart; say which you mean.
Stage 19 ran on one sentence: a set is a definite collection, decided by a clear membership rule. In this last lesson we carried that rule onto the number line, where the members fill a continuous band:
These are the exact tools a function will use in Stage 20 to announce its domain — the set of inputs it is allowed to eat. The bridge out of geometry is built; you are ready to cross it.
Write the set { x | −2 ≤ x < 4 } as an interval.
[−2, 4). The left end is closed (≤, so −2 is in → square bracket), the right end is open (<, so 4 is out → round bracket).
Is 5 ∈ (2, 5)? Explain.
No. The right end of (2, 5) is open, meaning 2 < x < 5, so 5 is excluded: 5 ∉ (2, 5). (It would be in (2, 5].)
Write "all real numbers greater than 3" as an interval.
(3, ∞). The 3 is excluded (strictly greater → open at 3), and ∞ is a direction, not a number, so its end is always open too.
Find [0, 4] ∩ (2, 6).
(2, 4]. Lower end = max(0, 2) = 2; only (2, 6) reaches 2 and it is open, so 2 is out. Upper end = min(4, 6) = 4; only [0, 4] reaches 4 and it is closed, so 4 is in.
Find [1, 4) ∪ [4, 7].
[1, 7]. The first stops just before 4, the second starts at 4 and 4 ∈ [4, 7], so the gap at 4 is filled and the two pieces join into one interval.
Write the neighborhood of radius 0.1 about 3, both as an interval and in absolute-value form.
(2.9, 3.1) = { x | |x − 3| < 0.1 }. Centre 3, radius 0.1, so the ends are 3 ∓ 0.1; both ends are hollow because |x − 3| < 0.1 is strict.
Six questions to lock it in. Tap the answer you think is right.
This lesson turns a solution set into a named object. The big idea is that a continuous range of real numbers is a set — you cannot list its members, so interval notation is how you write it down. Everything the student learned about ∈, ⊆, ∪, and ∩ in 19.1–19.4 reappears here on the number line, and the payoff arrives in Stage 20: a function's domain is almost always stated as an interval or a union of intervals.
The symbols are most of the difficulty, so insist students read them aloud: "[1, 5] is the closed interval from 1 to 5, including both ends"; "(a − δ, a + δ) is the set of all x such that the distance from x to a is less than δ." Three misconceptions are worth heading off directly. (1) Writing a closed bracket next to ∞ — there is no number there to include, so it is always "[2, ∞)", never "[2, ∞]". (2) Thinking (a, b) and [a, b] are the same set — they differ by exactly the two endpoints, and that difference is the whole point of open vs. closed. (3) Mishandling the shared endpoint when intersecting or uniting — drill that the stricter (open) bracket wins for ∩ and the looser (closed) for ∪. Finally, flag the notation clash head-on: the interval (1, 5) and the coordinate point (1, 5) look identical; only context decides.
Common Core: HS interval notation; A-REI.D and F-IF.B.5 (solution sets on the number line; domains expressed as intervals), and a deliberate bridge toward limits via the neighborhood (a − δ, a + δ).