Now that you can describe a set with a rule (\{x \mid x > 3\}), there's an even faster shorthand for one specific, extremely common kind of set: a stretch of the number line. That shorthand is called interval notation. You'll use it for the rest of this course every time an answer is "any number in this range" instead of one single value.
Think of an interval as a fence around a stretch of the number line. The brackets tell you whether the fence posts themselves count as "inside the yard."
A round parenthesis ( or ) is an open gate. Right up to that point, but not
touching it. A square bracket [ or ] is a solid fence post. That exact number
is included.
So (2, 5) is "everything strictly between 2 and 5." Get as close as you want to 2 or 5, but never actually land on them. [2, 5] is the same stretch plus the two endpoints themselves, fence posts and all.
Infinity always gets a round parenthesis, never a bracket. Here's the reasoning, not just the rule. A bracket claims "this exact value is included." But \infty isn't a number sitting at some spot on the line waiting to be included. It's a direction: "keeps going forever." You can't fence in a direction with a solid post, so it always gets the open, "keeps going" parenthesis.
One more habit is worth building now. The smaller number always goes on the left, the bigger on the right, no exceptions: (2, 5), never (5, 2). This isn't arbitrary styling either. Interval notation is shorthand for "walk along the number line starting here and stop there," and a number line only reads left to right (small to large). Writing (5, 2) would ask the reader to walk backwards, which isn't a convention anyone uses. So if a calculation ever hands you a bigger number first, that's your cue to swap the order before writing the final interval, not a sign something's broken.
An interval is the set of all real numbers between two endpoints. Interval notation uses parentheses and brackets to show whether each endpoint is included:
- Open interval (a, b): every number strictly between a and b; endpoints excluded. Equivalent to \{x \mid a < x < b\}.
- Closed interval [a, b]: every number between a and b, endpoints included. Equivalent to \{x \mid a \le x \le b\}.
- Half-open intervals [a, b) or (a, b]: one endpoint included, one excluded.
For an interval with no upper or lower bound, we use \infty (infinity) or -\infty (negative infinity). This always takes a parenthesis, never a bracket, since infinity isn't a number you can actually reach and "include": (a, \infty) = \{x \mid x > a\} \qquad (-\infty, b] = \{x \mid x \le b\}
Formula
Interval Notation
For endpoints a < b: (a,b)=\{x\mid a<x<b\} \qquad [a,b]=\{x\mid a\le x\le b\} [a,b)=\{x\mid a\le x<b\} \qquad (a,b]=\{x\mid a<x\le b\} (a,\infty)=\{x\mid x>a\} \qquad (-\infty,b]=\{x\mid x\le b\}
![A number line from 0 to 5 with a thick indigo segment from 1 to 4, and filled circles at both 1 and 4, showing the closed interval [1, 4]](/assets/images/number-systems/interval-closed.svg?v=1792993961)
Number line from -8 to 8 showing the interval (-2, 5]
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Check your understanding (counts toward your progress)
Graph the interval (-3, 3]: left excluded, right included.
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What this task checks
Structure checked automatically. Whether your ideas are convincing isn't graded, so reread with the checklist.
The compiler says
A model answer
Why
(-3,3] has an open circle at -3 (excluded) and a filled dot at 3 (included).
Correct answer
Also accepted:
Why
Check each endpoint separately: does the interval's symbol at that endpoint include or exclude it?
How many integers lie in the interval [2,7)?
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Why
The interval includes integers from 2 through 6 inclusive, which is 5 integer(s).
Correct answer
Also accepted:
Why
An open endpoint excludes that integer; a closed (bracketed) endpoint includes it.
Which interval correctly attaches its brackets — square on the left (included), round on the right (excluded)?
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This has it backwards — the round parenthesis is on the left (excluded) and the bracket is on the right (included), the opposite of what's asked.
Both sides are open here — neither endpoint is included at all.
Both sides are closed here — both endpoints are included, not just the left one.
Why
Each endpoint's symbol is independent: a square bracket [ on the left means 1 is included, and a round parenthesis ) on the right means 6 is excluded. [1, 6) is the only option matching "closed left, open right."
Check each option's left bracket and right bracket separately against 'included on the left, excluded on the right'.
Practice: Reading, graphing, and applying intervals (5 problems · doesn’t affect your progress)
What does the interval (2, 7) include?
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Round parentheses mean EXCLUDED endpoints. Brackets [2,7] would be the "including both" version.
Parentheses describe everything BETWEEN the endpoints, not the endpoints alone — this is a whole continuous stretch, not two isolated points.
That would be (7, \infty). This interval is bounded on both sides, between 2 and 7.
Why
Round parentheses signal excluded endpoints — (2,7) is every real number strictly between 2 and 7, getting arbitrarily close to either boundary but never touching it.
Every number strictly between 2 and 7 — not including 2 or 7 themselves
Why does infinity always get a parenthesis, never a bracket, in interval notation?
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It follows from what a bracket actually claims — "this value is included" — and infinity was never a landable value to begin with.
Bracket vs. parenthesis is about inclusion/exclusion of an endpoint, unrelated to whether a number is positive or negative.
Infinity can be either the upper bound (\infty) or the lower bound (-\infty, negative infinity) — its role isn't fixed.
Why
A bracket means "this exact number is part of the set." Infinity represents a direction that keeps going forever, not a specific number you could ever actually reach — so it always takes the open, excluded-endpoint parenthesis.
A bracket claims a specific value is included, but infinity isn't a value you can land on
What is [3, 3]?
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The number 3 itself IS at least 3 and at most 3 simultaneously — it satisfies both boundary conditions, so it belongs to this set.
There's no unbounded piece here — both endpoints are the fixed number 3, not infinity.
It's a perfectly valid, if unusual, closed interval — both brackets are legal, they just happen to name the same number.
Why
[3,3] asks for every number that is simultaneously \ge 3 and \le 3 — only 3 itself satisfies both. A closed interval with matching endpoints degenerates to a single point rather than an empty set.
A single-point set: just the number 3
Graph the interval [-4, 2]: both endpoints included.
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What this task checks
Structure checked automatically. Whether your ideas are convincing isn't graded, so reread with the checklist.
The compiler says
A model answer
Why
[-4,2] includes both endpoints — filled dots at -4 and 2, with the solid line between them.
Correct answer
Also accepted:
Why
Check whether each endpoint's own bracket style should be square or round.
A part is manufactured to spec "12 mm, plus or minus 0.5 mm" (both boundary measurements pass inspection). Which interval matches this tolerance?
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Open parentheses would reject a part measuring EXACTLY 11.5 mm or 12.5 mm — but the spec says both boundary measurements pass.
This rejects a part measuring exactly 12.5 mm, but the tolerance explicitly allows that boundary value.
This rejects a part measuring exactly 11.5 mm, but the tolerance explicitly allows that boundary value.
Why
"Plus or minus 0.5" from 12 gives boundaries of 11.5 and 12.5, and "both boundary measurements pass" means both endpoints are included — a closed interval, [11.5, 12.5].
Decide whether both boundary measurements should be included or excluded before choosing a bracket or parenthesis for each side.
[11.5, 12.5]
A common mistake: mixing up bracket direction
It's tempting to think of an interval as being "open" or "closed" as a single global setting, the way a door is either open or shut. It isn't. Each end has its own switch, flipped independently. [2, 5) has its left switch set to "closed" (2 is included) and its right switch set to "open" (5 is excluded). Nothing about the left switch's setting tells you anything about the right one. The fix for this mistake is mechanical, not conceptual. After writing an interval, check each endpoint one at a time: "does this specific number get included, yes or no?" Do that instead of trying to decide the interval's "openness" all at once.
A frequent error is attaching the wrong bracket to the wrong side, for instance writing (2, 5] when [2, 5) was intended. Each bracket is tied to a specific endpoint's inclusion, not to a general "openness" of the interval as a whole. The left symbol governs only the left (smaller) endpoint, and the right symbol governs only the right (larger) endpoint, independently of each other.
Why this matters outside of math class
Interval notation is really just a tidy way to write down "an acceptable range," and acceptable ranges are everywhere. A speed limit sign reading "45" is really saying "your speed must be in (-\infty, 45]" (any speed up to and including 45, with the lower bound left unspoken since negative speed isn't a real concern). A thermostat set to "between 68 and 72" describes the interval [68, 72]. A store coupon good "for orders over $50" describes (50, \infty). Notice the open parenthesis there: an order of exactly $50 usually doesn't qualify for a coupon that requires being strictly over that amount. That's precisely the open/closed distinction this section is about. Once you're fluent in interval notation, you'll start noticing it's the natural language for describing limits, thresholds, and acceptable ranges anywhere they appear, not just inside a math textbook.
A manufacturing tolerance is another good everyday closed interval. A part specified as "10 mm, plus or minus 0.2 mm" is really the interval [9.8, 10.2]: both boundary measurements are acceptable, not just the values strictly between them. That's why a quality inspector uses square brackets in their head (even without ever writing the notation down) when deciding whether a borderline part passes or fails.
Translating between interval, inequality, and picture
The same set of numbers can be written three ways: as an inequality, as an interval, and as a shaded picture of the number line. You should be able to move between all three instantly, in any direction.
Word problems rarely hand you an inequality symbol directly, though. They describe a range in English, and the first translation step is knowing which everyday phrase maps to which symbol.
Two of these are easy to mix up because they sound similar but point in opposite directions:
- "At least" means "this value or bigger." A concert that's "18 and over," at least 18, lets an 18-year-old in: x \ge 18.
- "At most" means "this value or smaller." A carry-on bag allowed "at most 22 inches" lets a bag that's exactly 22 inches through: x \le 22.
"No more than" and "no less than" are just "at most" and "at least" said as a double negative. "No more than 3 items" still means x \le 3, boundary included.
The moment a phrase drops the "or equal to" idea, like "more than," "greater than," or "fewer than," the boundary itself stops qualifying. The interval needs a parenthesis instead of a bracket at that endpoint.
| Phrase | Inequality | Endpoint |
|---|---|---|
| "at least a" | x \ge a | included |
| "at most a" | x \le a | included |
| "no more than a" | x \le a | included |
| "no less than a" | x \ge a | included |
| "more than a" / "greater than a" | x > a | excluded |
| "fewer than a" / "less than a" | x < a | excluded |
A phrase that carries the sense of "or equal to" ("at least," "at most," "no more than," "no less than") keeps the boundary value, so it gets a bracket. A phrase that's strictly one-sided ("more than," "greater than," "fewer than," "less than") drops the boundary value, so it gets a parenthesis.
Worked example
Inequality → interval → picture
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1
Start with the inequality -1 \le x < 4 — this describes every number from -1 up to (but not including) 4.
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2
The left boundary, -1, is included (the \le has an "or equal to"), so it gets a square bracket: [-1, \ldots
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3
The right boundary, 4, is excluded (strict <, no "or equal to"), so it gets a round parenthesis: \ldots, 4)
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4
Full interval: [-1, 4). On the number line, that's a filled-in dot at -1, an open circle at 4, and a solid line connecting them.
Worked example
Starting from interval notation and working backward
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1
This time, go the reverse direction: you're handed the interval (-6, 0], and asked for the matching inequality and picture.
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2
Look at the left symbol: a round parenthesis at -6. That means "strictly greater than," excluded: -6 < x.
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3
Look at the right symbol: a square bracket at 0. That means "up to and including," included: x \le 0.
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4
Combine the two halves into one compound inequality: -6 < x \le 0. On the number line: an open circle at -6, a solid line moving right, and a filled-in dot at 0 where the line stops. Reading the interval and reading the inequality are two views of the exact same information. Neither one is "more correct" than the other; they're just different notations for an identical set of numbers.
Check your understanding (counts toward your progress)
The inequality is x>0 and x\le5. Graph the matching interval.
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What this task checks
Structure checked automatically. Whether your ideas are convincing isn't graded, so reread with the checklist.
The compiler says
A model answer
Why
Strict > at 0 means excluded (open circle); \le at 5 means included (filled dot): (0,5].
Correct answer
Also accepted:
Why
Translate each inequality symbol into the matching bracket or parenthesis for its own endpoint.
Which inequality matches the interval [-6, 0]?
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Both brackets here are square (closed), meaning both endpoints ARE included — strict inequalities on both sides would exclude them.
The left bracket is also square (closed) — -6 should be included with \le, not excluded with strict <.
The right bracket is also square (closed) — 0 should be included with \le, not excluded with strict <.
Why
Both endpoints of [-6, 0] use square brackets, meaning both are included: -6 \le x and x \le 0, combined as -6 \le x \le 0.
Check whether each bracket is square or round, then match that to whether each inequality should be strict or 'or equal to'.
Which interval notation matches -23\le x\le 6 ?
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Why
A strict inequality (<) uses a parenthesis (excluded endpoint); a \le uses a bracket (included endpoint) -- matching each side of -23\le x\le 6 gives [-23,6].
Practice: Translating between inequality, interval, and picture (5 problems · doesn’t affect your progress)
On a number line picture, an OPEN circle at an endpoint corresponds to which bracket style?
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A filled/closed circle matches the bracket (included). An open circle is the parenthesis — excluded.
Open and closed circles are exactly how interval endpoints get drawn on a number line — they translate directly to parentheses/brackets.
The sign of the number has nothing to do with it — it's purely about whether that boundary value itself is part of the set.
Why
An open (unfilled) circle means that exact point is NOT part of the set — matching a round parenthesis. A filled/closed circle means the point IS included — matching a square bracket.
Round parenthesis ( or ) — endpoint excluded
The inequality is x\ge3 and x<8. Graph the matching interval.
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What this task checks
Structure checked automatically. Whether your ideas are convincing isn't graded, so reread with the checklist.
The compiler says
A model answer
Why
x\ge3 includes 3 (filled dot); x<8 excludes 8 (open circle): [3,8).
Correct answer
Also accepted:
Why
Write x < 8 in interval notation.
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A bracket at 8 would include 8 — but strict < means 8 itself is excluded.
This describes "greater than or equal to 8," the opposite direction from "less than 8."
This describes "greater than 8," the opposite direction, and also wrongly excludes numbers less than 8.
Why
x < 8 has no lower bound (extends to -\infty) and excludes 8 itself (strict inequality, so a parenthesis): (-\infty, 8).
Decide whether 8 itself belongs in the set, and note that there's no lower boundary given at all.
(-\infty, 8)
Which inequality describes the interval (-3, 6]?
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That drops the closed bracket at 6 — (-3,6] includes 6, but this inequality excludes it.
The parenthesis at -3 means it's excluded — but this inequality includes it with \le. Only the right side should have "or equal to."
This has the inclusion on the wrong side — (-3, 6] excludes -3 (parenthesis) and includes 6 (bracket), the reverse of what's written here.
Why
The parenthesis at -3 means "strictly greater than" (-3 < x); the bracket at 6 means "up to and including" (x \le 6). Together: -3 < x \le 6.
Look at each endpoint's symbol separately — parenthesis means strict inequality, bracket means 'or equal to'.
-3 < x \le 6
A student translates "x is at most 5" as (5, \infty). What went wrong?
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"At most 5" means 5 and everything below it — this interval instead describes everything ABOVE 5, excluding 5 itself. Two separate errors, not a correct answer.
The direction is also wrong: "at most 5" points toward smaller numbers (-\infty), not toward \infty.
The bracket is also wrong: "at most" includes the boundary value, so 5 needs a bracket, not a parenthesis.
Why
"At most 5" means x \le 5 — everything from -\infty up to and including 5. The correct interval is (-\infty, 5]: the student had both the direction and the endpoint inclusion backwards.
Work out separately which direction 'at most' points, and whether the boundary value itself should be included.
Translate 'at most 5' into an inequality first (x \le 5), then convert that inequality into interval notation.
The direction is backwards (should point toward smaller numbers) AND 5 should be included, not excluded
Combining intervals with \cup
Sometimes the answer to a problem isn't one continuous stretch — it's two separate pieces. That's what the union symbol from the last section was built for.
This is the exact same \cup from the sets section, "pour both groups into one." It's just applied to two stretches of the number line instead of two lists of numbers. Written out, (-\infty, -2) \cup (5, \infty) reads as "everything way off to the left of -2, combined with everything way off to the right of 5." Think of it as two separate rooms in the same house, connected by a hallway made of the word "or." Notice there's a gap in the middle, from -2 to 5, that belongs to neither piece. That's what makes this a union of two intervals instead of one continuous one.
When a solution set consists of two or more separate intervals, they are combined with the union symbol \cup. For example, "every number less than -2 or greater than 5" is written: (-\infty, -2) \cup (5, \infty)
Worked example
A trickier union — three pieces, and an overlap check
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1
Describe: "every number less than 0, or between 2 and 4 (inclusive), or greater than 10."
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2
Three separate conditions, joined by "or," means three intervals joined by \cup. Take them one at a time: "less than 0" is (-\infty, 0).
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3
"Between 2 and 4, inclusive" (both endpoints included) is [2, 4]. "Greater than 10" is (10, \infty).
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4
Join all three with \cup: (-\infty, 0) \cup [2, 4] \cup (10, \infty) Before finalizing, it's worth a quick sanity check: do any of these three pieces overlap? Look at the boundary near 0 and 2 first: (-\infty,0) stops before 0, and [2,4] starts at 2, so there's a gap between them (everything from 0 to 2). Now the boundary near 4 and 10: [2,4] stops at 4, and (10,\infty) starts after 10, so there's another gap (4 to 10). No overlaps, so all three pieces stay separate, and none of them can be merged into a bigger single interval.
When intervals secretly merge into one
Watch what happens with (-\infty, 3] \cup (1, \infty). These look like two separate pieces at first glance, the same way (-\infty,-2)\cup(5,\infty) was. But check for overlap first. Does anything satisfy both "\le 3" and "> 1"? Sure: 2, for instance. The two pieces overlap in the middle, which means there's no gap at all between "everything up to 3" and "everything past 1." Together they cover every real number. The "union" here quietly collapses to (-\infty, \infty), all real numbers. The lesson: always check for a gap before locking in a multi-piece answer. A union is only two rooms connected by a hallway if there's an actual wall, a gap, between them. Overlapping pieces just describe one bigger room.
Two intervals combined with \cup only remain a genuine union of two separate pieces if they don't touch or overlap. If they share numbers, or their endpoints coincide at a point where one is included, the "union" simplifies to a single continuous interval. It's still technically true to write it as a union, but it's not in simplest form.
Build (-\infty, 3] \cup (1, \infty) yourself and watch the result row merge into one piece the moment the two overlap:
Check your understanding (counts toward your progress)
Build the union below in the result row, then check whether it stays two separate pieces or merges into one.
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What this task checks
Structure checked automatically. Whether your ideas are convincing isn't graded, so reread with the checklist.
The compiler says
A model answer
Why
[-8,3] and (1,8] overlap between 1 and 3, so there's no gap between the two pieces. The union collapses into the single interval [-8, 8].
Correct answer
Also accepted:
Why
Drag both pieces into place first, then look for a genuine gap between them before deciding whether the result stays two pieces or merges into one.
Worked example
A merge that doesn't collapse all the way to every real number
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1
Determine whether [1, 5] \cup (3, 8) is truly two separate pieces, and rewrite it in simplest form if it isn't. This time, unlike the last example, the two pieces are both bounded on both ends.
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2
Check for overlap first: is there a number satisfying both "1 \le x \le 5" and "3 < x < 8"? Yes — 4, for instance, passes both tests. The two pieces overlap, so this isn't a genuine two-piece union.
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3
Since they overlap, the combined set is one continuous stretch. Find its leftmost and rightmost boundary separately: the leftmost point of either piece is 1 (from [1,5], included, since no part of (3,8) reaches that far left); the rightmost point is 8 (from (3,8), excluded, since [1,5] stops at 5).
-
4[1, 5] \cup (3, 8) = [1, 8)
Same collapsing idea as (-\infty, 3] \cup (1, \infty) above, just on a smaller, bounded scale: it doesn't have to swallow the whole real line to still be a single merged interval instead of two.
Check your understanding (counts toward your progress)
What is (-\infty, -1) \cap (3, \infty) — the INTERSECTION instead of the union?
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That's the union, describing the exact opposite operation (everything satisfying EITHER condition, not both).
This is the interval BETWEEN -1 and 3, but intersection with two conditions like these has nothing to do with what's between them.
This would only happen if the two pieces overlapped completely — here they don't overlap at all.
Why
No real number can be simultaneously less than -1 AND greater than 3 — those requirements contradict each other. The intersection of two conditions with no overlap is the empty set, \varnothing.
Ask whether any single number could satisfy both 'less than -1' and 'greater than 3' at the same time.
How many separate pieces does (-\infty,3]\cup(1,\infty) simplify to, once you check for overlap?
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Why
A number like 2 satisfies both "\le3" and ">1" — the pieces overlap and cover every real number, collapsing to one piece: (-\infty,\infty).
Correct answer
Also accepted:
Why
Pick a test number and check whether it's covered by at least one of the two pieces, then ask whether every real number passes.
Practice: Unions of intervals, and when they merge (4 problems · doesn’t affect your progress)
Over the range -6\le x\le8, how many integers satisfy (-\infty,-2)\cup(5,\infty)?
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Why
Among -6,\ldots,8: x<-2 gives -6,-5,-4,-3 (four values), and x>5 gives 6,7,8 (three values) — seven total.
Correct answer
Also accepted:
Why
"Every number less than -1 or greater than 3" in interval notation is:
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That describes numbers BETWEEN -1 and 3 — the opposite of "less than -1 or greater than 3," which is everything OUTSIDE that range.
Intersection (\cap) would demand a number satisfy BOTH conditions at once — but nothing is simultaneously less than -1 AND greater than 3. Union (\cup) is what "or" calls for.
This is a single closed interval between -1 and 3 — the described set is two separate unbounded pieces, not one bounded piece.
Why
"Or" between two separate conditions calls for union: everything satisfying the first piece, combined with everything satisfying the second. (-\infty,-1) covers "less than -1," (3,\infty) covers "greater than 3," joined with \cup.
Match the word 'or' to the correct set operation between the two pieces.
(-\infty, -1) \cup (3, \infty)
(-\infty, 3] \cup (1, \infty) looks like a union of two separate pieces. What does it actually simplify to?
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Check for overlap first: a number like 2 satisfies BOTH "\le 3" and "> 1" — the pieces aren't actually separate, they overlap.
Union never "cancels" pieces to nothing — it only ever combines or grows, it can't produce the empty set from two nonempty pieces.
That would be the INTERSECTION of the two pieces, not their union — union keeps everything from either piece, not just the overlap.
Why
Any real number is either >1 (covered by the second piece) or \le 1, and if \le 1 then it's also \le 3 (covered by the first piece) — so every real number is covered by at least one piece. With no gap between them, the union collapses to all real numbers, (-\infty,\infty).
Pick a test number like 2 and check whether it's covered by at least one of the two pieces — then ask whether every real number passes that same test.
Consider any number \le 1 — is it also covered by the first piece, (-\infty, 3]?
(-\infty, \infty) — all real numbers, because the two pieces overlap and leave no gap
(-\infty,2)\cup(2,\infty) — what single description matches this union?
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2 itself is excluded from BOTH pieces (each uses a parenthesis at 2) — it's not truly all real numbers.
This is the opposite — 2 is specifically the ONE number left out, not the only one included.
Plenty of numbers satisfy at least one piece — this union is far from empty.
Why
Every real number except 2 satisfies at least one of the two pieces (less than 2, or greater than 2); 2 itself satisfies neither, since both pieces use a parenthesis there.
Pick a handful of test numbers (less than 2, greater than 2, and 2 itself) and check which piece(s) each one satisfies.
All real numbers except 2
An edge case worth seeing once: a one-point interval
What happens if both endpoints of a closed interval are the same number, like [3, 3]? It looks unusual, but the rule from the very top of this section still applies without any exception: "every number between 3 and 3, inclusive." The only number that's simultaneously "at least 3" and "at most 3" is 3 itself. So [3,3] is a perfectly valid interval that happens to describe a single point, not a stretch at all. Compare this to (3, 3) (the open version), which asks for a number strictly between 3 and 3, and there isn't one, so (3,3) = \varnothing, the empty set. Same two numbers, different brackets. The difference between "a single valid point" and "nothing at all" comes down entirely to whether the endpoints are included: the exact distinction this whole section has been building toward.
Why this matters
Interval notation is the standard answer format for every inequality you'll solve in the next chapter, and it's how we'll describe the domain and range of a function a few chapters from now. Get comfortable translating between inequality, interval, and picture now, on easy numbers, and it will be one less thing to think about when the equations get harder.
It's also worth noticing the pattern this section shares with the last one. Every idea here (open vs. closed, union of pieces, even the odd one-point edge case) is really just the set vocabulary from the previous section (\in, \subset, \cup, \varnothing) wearing number-line clothes. Nothing new is being invented; the same small toolkit keeps getting reapplied to a new kind of set. That pattern, old tools applied to new context, is going to repeat constantly for the rest of this course. Noticing it explicitly now makes each new chapter feel like less new material than it might otherwise seem.
Where this shows up
Medicine & Pharmacology
A drug's prescribing label often states a therapeutic window: the dose range where the medication is effective without becoming toxic. A window described as "150 to 300 mg per day" is the closed interval [150, 300]. Both boundary doses are still safe and effective, so the brackets are square, not round. Below 150 mg, the drug may not work; above 300 mg, side effects become dangerous. A pharmacist reasoning about a patient's dosage is, whether they'd phrase it this way or not, checking whether a number lies inside [150, 300]. That's the endpoint-by-endpoint test this section teaches. Get the brackets wrong on paper and it's a graded problem. Get them wrong on a real prescription and the interval's open/closed distinction is the difference between a safe dose and an overdose.
What's next
Every interval above assumed you already knew what kind of number could fill it. The next section names those kinds precisely: naturals, integers, rationals, irrationals. That way a statement like "x is a rational number in [0,5]" means something specific, not just "some number in this range."