Interval Notation
There is no way to reason your way from "3 is included" to "use a square bracket" - it is a convention. What makes it stick is seeing all three notations move together.
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Interval notation is not difficult. It is arbitrary, which is worse. There is no argument that gets you from "3 belongs to this set" to "so use a square bracket" - somebody chose it, and now you have to know it.
What actually makes it stick is watching the three ways of writing a set move at the same time. Drag an end below, then switch it between included and excluded.
Move an end, then switch it between included and excluded. The bracket changes shape and the dot fills in - they are two spellings of one fact.
The two brackets
Every interval is written smallest end first, largest second, and each end carries one of two marks:
| you mean | inequality | notation | on the line |
|---|---|---|---|
| the end is in | x <= 7 | 7] | filled dot |
| the end is out | x < 7 | 7) | hollow dot |
That is the whole convention. Everything else is bookkeeping.
So -2 <= x < 5 - two and a bit ideas in one line - becomes
[-2, 5). The square bracket on the left says -2 is in the set; the
parenthesis on the right says 5 is not.
A useful sanity check: the bracket shape points the same way as the sign. A square bracket has a flat, closed end - like the flat line under the ≤ sign. A parenthesis is open and rounded, like the plain <.
Infinity is always open
x > 3 has no upper end, so you write (3, ∞).
Notice the parenthesis on the right. It is not a stylistic choice - it is
forced. A square bracket means "this endpoint is in the set", and infinity is
not a number, so it cannot be in anything. Writing [3, ∞] is claiming that
some value is infinity, which is why every textbook marks it wrong.
The same on the other side: x <= 5 is (-∞, 5] - open against the infinity, closed against the 5.
Two stretches at once
Some sets are not one piece. "Everything below -2 or above 5" cannot be written as one interval, because the numbers in between are not in it. Join the pieces with a union:
(-∞, -2) ∪ (5, ∞)
Always write the left-hand piece first, and never write it backwards as
[5, -2]. An interval's first number is the smaller one, so a backwards
interval describes nothing at all.
This shape turns up most often in domains. The domain of 1/(x-3) is every number except 3, which as intervals is (-∞, 3) ∪ (3, ∞) - the number line with one point punched out.
From an inequality to an interval
Most exam questions hand you an inequality to solve and then want the answer as an interval, so the two halves matter equally. Solve first:
The answer is x < 5. There is no lower end, and 5 is excluded, so in interval notation that is (-∞, 5).
One trap while you solve: multiplying or dividing by a negative flips the sign. -3x <= 12 becomes x >= -4, not x <= -4 - and if you miss the flip, you get an interval pointing in exactly the wrong direction.
Try one
Then say the answer out loud as an interval before you move on. Being able to solve it and being able to write it are two separate skills, and the second one is where the marks are lost.
Common questions
- What is interval notation?
- A short way of writing a stretch of the number line. You give the two ends in order, smallest first, and the type of bracket says whether each end is itself included: a square bracket includes it, a round parenthesis excludes it. So [2, 7) means everything from 2 up to but not including 7.
- When do I use brackets and when parentheses?
- A square bracket for an endpoint that belongs to the set, a parenthesis for one that does not. It matches the inequality sign directly: less than or equal to gets a bracket, strictly less than gets a parenthesis. On a number line the same distinction is a filled dot against a hollow one.
- Why does infinity always get a parenthesis?
- Because infinity is not a number, so nothing can ever be equal to it. An endpoint gets a square bracket only when the set contains that value, and no set contains infinity. So it is always (-∞, 5] or [3, ∞) - never a square bracket against the infinity symbol.
- How do I write two separate stretches?
- Join them with the union symbol ∪, writing the left-hand stretch first: (-∞, -2) ∪ (5, ∞) is everything below -2 together with everything above 5. This is what an inequality with 'or' looks like in interval notation, and it comes up constantly for domains where a single value has to be excluded.
- What is the difference between interval notation and set-builder notation?
- They describe the same sets in different styles. Interval notation names the two ends and lets brackets carry the detail; set-builder notation writes the condition out, as in the set of x such that 2 is at most x and x is under 7. Interval notation is shorter and is what most courses expect for domains and ranges.
Now do it yourself, properly
Reading someone else's working is not the same as being able to do it. The Coddy math course puts you on a board that checks every move you make, so you find out where you actually stand.
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