Think of directions for a stretch of road. You can say which mile markers a driver may pass, state a condition for an allowed position, or write the beginning and ending markers. These are three descriptions of the same road. Inequality notation states the comparisons. Set-builder notation wraps the condition in braces and reads it as a collection. Interval notation writes the two limits. You must also say whether each boundary is part of the road. A filled dot means you may stand there; a hollow dot means you may get close but that exact position is excluded.
The union includes 1 through 3 and values strictly above 5, leaving the gap unshaded.
Reminder
Compound conditions. −2 ≤ x < 3 means x ≥ −2 and x < 3; both must hold.
Order of numbers. −4 < −1, so −4 is written first in an interval.
Why it works. The notations are conventions chosen to record the same membership decisions compactly. A square bracket includes its endpoint; a parenthesis excludes it. Ignoring that distinction changes whether an input is allowed. Infinity is a direction of unlimited continuation, not a real endpoint you can reach, so it always takes a parenthesis. Writing the smaller limit first matches the number line's order. A union combines membership in either set, including any overlap. Without it, writing only the outside limits could accidentally fill an excluded gap.
RuleUse [ or ] for an included endpoint, ( or ) for an excluded endpoint or infinity. Write intervals from smaller to larger; union (∪) means or, including membership in both sets.
The same idea, five ways
Say it
Say at least negative two and less than three.
Write it
The set includes −2, excludes 3, and contains every real number between them.
In math
−2 ≤ x < 3
{x | −2 ≤ x < 3}
[−2, 3)
Graph words: closed dot at −2, open dot at 3, shade between
Like
A road segment where one boundary post belongs to your route and the other does not.
See itThe notation and the endpoint dots describe identical membership.
The same idea, other ways
As a fence with gates
An interval gives the stretch between two fence posts. A square bracket keeps a post in your property; a parenthesis leaves the post outside. A missing stretch needs two intervals rather than one.
As a membership question
For [−2, 3), ask whether each candidate satisfies −2 ≤ x < 3. This includes −2 and excludes 3. Set-builder notation records exactly that question.
As three translations
x ≤ 7, {x | x ≤ 7}, and (−∞, 7] all include 7 and every smaller real number. The descriptions change, but the allowed members do not.
Inequality notation
Set-builder notation
Interval notation
x > 5
{x | x > 5}
(5, ∞)
−2 ≤ x < 3
{x | −2 ≤ x < 3}
[−2, 3)
x ≤ 7
{x | x ≤ 7}
(−∞, 7]
1 ≤ x ≤ 3 or x > 5
{x | 1 ≤ x ≤ 3 or x > 5}
[1, 3] ∪ (5, ∞)
.1Inequality notation
Inequality notation compares a candidate number with its limits. The word and means the candidate must pass both comparisons. The word or means one comparison or the other is enough. Do not turn two separate stretches into a single and statement.
A strict < or > comparison excludes its boundary.
A ≤ or ≥ comparison includes its boundary.
A compound inequality such as −2 ≤ x < 3 means both limits hold.
A strict comparison begins after 5.
Reminder
Compound conditions. −2 ≤ x < 3 means x ≥ −2 and x < 3; both must hold.
Order of numbers. −4 < −1, so −4 is written first in an interval.
The same idea, five ways
Say it
Say strictly greater than five.
Write it
The number must exceed 5, so equality is excluded.
In math
x > 5
{x | x > 5}
(5, ∞)
Graph words: open dot at 5, shade right
Like
An entry sign admits only positions passing the stated height checks.
See itA strict comparison begins after 5.
Worked exampleNotation ladder 1: one open ray
Write x > 5 in set-builder and interval notation.
Interval: (5, ∞). The endpoint symbols record which limits belong.
We need a different description with exactly the same allowed members.State what is being found before choosing the calculation.
Read > as strictly greater than, so 5 is excluded.The inequality has no equality part.
Write {x | x > 5}.The braces name a set and the bar means such that.
Write (5, ∞).The set starts beyond 5 and has no upper endpoint; both symbols use parentheses.
Answer
Set-builder: {x | x > 5}.
Interval: (5, ∞).
Check 6 belongs to every description; 5 and 4 belong to none.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: x > 5 includes 5.
Strictly greater excludes equality.
✓ Instead: Keep only inputs strictly above 5, described by (5, ∞).
Tips and tricks
Read the comparison aloud before choosing an endpoint symbol.
.2Set-builder notation
Set-builder notation is a sentence compressed into symbols. Braces, { and }, mean the set of. The vertical bar means such that. After the bar comes the condition an element must satisfy. Here x is understood to be a real number unless another kind of input is specified.
{x | x > 5} reads the set of x such that x is greater than 5.
A condition may use words as well as inequalities.
A listed set {2, 7} names its elements; a set-builder condition explains membership.
{x | x ≤ 7}
the set of real x
such that x is at most 7
Read the bar as such that, not as a fraction bar.
Reminder
Compound conditions. −2 ≤ x < 3 means x ≥ −2 and x < 3; both must hold.
Order of numbers. −4 < −1, so −4 is written first in an interval.
The same idea, five ways
Say it
Say the set of real x such that x is at most seven.
Write it
Braces name a set and the bar introduces its membership condition.
In math
x ≤ 7
{x | x ≤ 7}
(−∞, 7]
Graph words: closed dot at 7, shade left
Like
A guest-list rule states who qualifies without listing every possible guest.
See it
{x | x ≤ 7}
the set of real x
such that x is at most 7
Read the bar as such that, not as a fraction bar.
Worked exampleA closed ray extending to the left
Write x ≤ 7 in set-builder and interval notation.
Interval: (−∞, 7]. The endpoint symbols record which limits belong.
We need a different description with exactly the same allowed members.State what is being found before choosing the calculation.
Include 7 and all smaller real values.The ≤ symbol includes equality.
Write {x | x ≤ 7}.Set-builder notation names the same membership condition.
Write (−∞, 7].The smaller end comes first, and the finite endpoint is included.
Answer
Set-builder: {x | x ≤ 7}.
Interval: (−∞, 7].
Check 7 and 6.9 belong, while 7.1 does not; there is no smallest real number in this set.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Read the bar in {x | x ≤ 7} as division.
Inside set-builder notation this bar introduces the membership condition.
✓ Instead: Read the set of real x such that x is at most 7.
Tips and tricks
Say the set of at the braces and such that at the bar.
.3Interval notation
Interval notation names a continuous stretch of real numbers by its endpoints. It includes all numbers between those limits, not only integers. Use it for a stretch, but use a list for isolated values. An interval may run without an upper or lower bound.
Inclusive means included; exclusive means excluded.
A bracket is square, while a parenthesis is curved.
[c, c] is a one-element set; (c, c) has no elements because no number lies strictly between c and itself.
Every real position between the limits belongs, including fractions and decimals.
Reminder
Compound conditions. −2 ≤ x < 3 means x ≥ −2 and x < 3; both must hold.
Order of numbers. −4 < −1, so −4 is written first in an interval.
The same idea, five ways
Say it
Say at least negative two and less than three.
Write it
The interval contains every real number between −2 and 3, with only the left endpoint included.
In math
−2 ≤ x < 3
{x | −2 ≤ x < 3}
[−2, 3)
Graph words: closed at −2, open at 3, shade between
Like
Two mile markers describe an unbroken permitted stretch of road.
See itEvery real position between the limits belongs, including fractions and decimals.
Worked exampleNotation ladder 2: one endpoint in, one out
Write −2 ≤ x < 3 in words, set-builder notation and interval notation.
Interval: [−2, 3). The endpoint symbols record which limits belong.
We need a different description with exactly the same allowed members.State what is being found before choosing the calculation.
Include −2 and exclude 3.The left comparison has equality, while the right comparison is strict.
Say all real numbers at least −2 and less than 3.Both bounds must hold at the same time.
Write {x | −2 ≤ x < 3} and [−2, 3).The left bracket includes −2 and the right parenthesis excludes 3.
Answer
Words: at least −2 and less than 3.
Set-builder: {x | −2 ≤ x < 3}.
Interval: [−2, 3).
Check The values −2 and 2.5 belong; 3 does not. All three descriptions give these same decisions.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: The finite set {−2, 3} equals [−2, 3].
The interval contains every real number between those limits, while the set lists only two numbers.
✓ Instead: Use {−2, 3} for two isolated members, and [−2, 3] for the full stretch.
Tips and tricks
Ask whether intervening fractions should belong.
.4Union of sets
Union means gather everything belonging to one collection or the other, or both. Picture pouring two trays of labeled keys onto one table. Keep every different label once. The collections may be separated, overlap, or contain only a few isolated members.
The symbol ∪ joins sets without inventing members.
Shared members appear once.
An intersection keeps only members in both sets. It explains why an input must pass all arithmetic restrictions, while union combines allowed pieces.
{2, 4, 6} ∪ {4, 7}
= {2, 4, 6, 7}
The shared member 4 is listed once
Union keeps membership from either set, including their overlap.
Reminder
Compound conditions. −2 ≤ x < 3 means x ≥ −2 and x < 3; both must hold.
Order of numbers. −4 < −1, so −4 is written first in an interval.
The same idea, five ways
Say it
Say in either set or in both.
Write it
Union collects all members from both sets, listing shared members once.
In math
{2, 4, 6} ∪ {4, 7} = {2, 4, 6, 7}
{x | x = 2 or x = 4 or x = 6 or x = 7}
Like
Combine two guest lists and keep each different guest once.
See it
{2, 4, 6} ∪ {4, 7}
= {2, 4, 6, 7}
The shared member 4 is listed once
Union keeps membership from either set, including their overlap.
Worked exampleUnion can combine overlapping finite sets
Find {2, 4, 6} ∪ {4, 7}.
{2, 4, 6, 7}.
These lines record the calculated results and their boundary choices.
We are collecting all members that belong to either set, keeping repeated members once.Naming the requested values tells us which taught method to use.
Collect members belonging to either set: 2, 4, 6, 4, 7.Union means the first set or the second set or both.
List 4 once and write {2, 4, 6, 7}.A set records whether an element belongs, not how many times it appears.
Answer
{2, 4, 6, 7}.
Check Each original set is contained in the answer. Listing {7, 2, 6, 4} would name the same set because a finite set has no required ordering.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: A shared member is listed twice in the union.
A set records membership rather than repeated appearances.
✓ Instead: {2, 4, 6} ∪ {4, 7} = {2, 4, 6, 7}.
Tips and tricks
Union keeps either collection's members and removes only repeated listings.
Strategy: step by step
1. Identify the shaded or allowed portion of the number line.
2. Use the closed dot / open dot convention: a filled dot is included and a hollow dot is excluded.
3. State comparisons with <, >, ≤ or ≥ for inequality notation.
4. Put the condition after the bar in {x | condition} for set-builder notation.
5. Write the lower limit first and the upper limit second in interval notation.
6. For separate pieces, join conditions with or and intervals with ∪. Keep every gap that is actually excluded.
Strategy
Translate an allowed set
1
Is a finite endpoint included?
YesUse a square bracket.
NoUse a parenthesis.
↓
2
Does the interval continue without a finite endpoint?
YesUse −∞ or ∞ with a parenthesis.
NoWrite the actual finite limit.
↓
3
Are there separate allowed pieces?
YesJoin their intervals with ∪ and their conditions with or.
NoWrite one interval without adding a gap.
1. Identify the shaded or allowed portion of the number line.
2. Read a filled dot as included and a hollow dot as excluded.
3. State comparisons with <, >, ≤ or ≥ for inequality notation.
4. Put the condition after the bar in {x | condition} for set-builder notation.
5. Write the lower limit first and the upper limit second in interval notation.
6. For separate pieces, join conditions with or and intervals with ∪. Keep every gap that is actually excluded.
Worked exampleTextbook: read two shaded intervals
A number line shades 1 through 3 with filled endpoints, and everything to the right of 5 with a hollow endpoint at 5. Describe the set three ways.
The union includes 1 through 3 and values strictly above 5, leaving the gap unshaded.
We are expressing the same allowed values in the requested notation.Naming the requested values tells us which taught method to use.
For the first shaded piece, write 1 ≤ x ≤ 3.Both endpoints have filled dots, so both are included.
For the second piece, write x > 5.The dot at 5 is hollow and the shading continues to the right without a last number.
Join the conditions with or and the intervals with ∪.Membership in either piece is enough; the unshaded gap is not included.
Read the braces and bar in {x | 1 ≤ x ≤ 3 or x > 5}.This says the set of real x such that one of the two stated conditions holds.
Answer
Inequality: 1 ≤ x ≤ 3 or x > 5.
Set-builder: {x | 1 ≤ x ≤ 3 or x > 5}.
Interval: [1, 3] ∪ (5, ∞).
Check 2 is included by the first piece and 6 by the second. Neither 4 nor 5 is included, so the gap remains.
Ladder: from easy to exam-hard. Press Try it first on any rung to hide its steps and use them as hints.
Rung 1Notation ladder 1: one open ray
Write x > 5 in set-builder and interval notation.
Interval: (5, ∞). The endpoint symbols record which limits belong.
We need a different description with exactly the same allowed members.State what is being found before choosing the calculation.
Read > as strictly greater than, so 5 is excluded.The inequality has no equality part.
Write {x | x > 5}.The braces name a set and the bar means such that.
Write (5, ∞).The set starts beyond 5 and has no upper endpoint; both symbols use parentheses.
Answer
Set-builder: {x | x > 5}.
Interval: (5, ∞).
Check 6 belongs to every description; 5 and 4 belong to none.
Rung 2Notation ladder 2: one endpoint in, one out
Write −2 ≤ x < 3 in words, set-builder notation and interval notation.
Interval: [−2, 3). The endpoint symbols record which limits belong.
We need a different description with exactly the same allowed members.State what is being found before choosing the calculation.
Include −2 and exclude 3.The left comparison has equality, while the right comparison is strict.
Say all real numbers at least −2 and less than 3.Both bounds must hold at the same time.
Write {x | −2 ≤ x < 3} and [−2, 3).The left bracket includes −2 and the right parenthesis excludes 3.
Answer
Words: at least −2 and less than 3.
Set-builder: {x | −2 ≤ x < 3}.
Interval: [−2, 3).
Check The values −2 and 2.5 belong; 3 does not. All three descriptions give these same decisions.
Rung 3Textbook: read two shaded intervals
A number line shades 1 through 3 with filled endpoints, and everything to the right of 5 with a hollow endpoint at 5. Describe the set three ways.
The union includes 1 through 3 and values strictly above 5, leaving the gap unshaded.
We are expressing the same allowed values in the requested notation.Naming the requested values tells us which taught method to use.
For the first shaded piece, write 1 ≤ x ≤ 3.Both endpoints have filled dots, so both are included.
For the second piece, write x > 5.The dot at 5 is hollow and the shading continues to the right without a last number.
Join the conditions with or and the intervals with ∪.Membership in either piece is enough; the unshaded gap is not included.
Read the braces and bar in {x | 1 ≤ x ≤ 3 or x > 5}.This says the set of real x such that one of the two stated conditions holds.
Answer
Inequality: 1 ≤ x ≤ 3 or x > 5.
Set-builder: {x | 1 ≤ x ≤ 3 or x > 5}.
Interval: [1, 3] ∪ (5, ∞).
Check 2 is included by the first piece and 6 by the second. Neither 4 nor 5 is included, so the gap remains.
Rung 4Notation ladder 4: every real number except one
Write x ≠ 2 using inequalities, set-builder notation and intervals.
Interval: (−∞, 2) ∪ (2, ∞). The endpoint symbols record which limits belong.
We need a different description with exactly the same allowed members.State what is being found before choosing the calculation.
Split the allowed inputs into x < 2 or x > 2.Every real number except 2 lies on one side of 2.
Write {x | x ≠ 2}.The condition says to omit that single member.
Write (−∞, 2) ∪ (2, ∞).Neither interval includes 2, and their union contains everything else.
Answer
Inequality: x < 2 or x > 2.
Set-builder: {x | x ≠ 2}.
Interval: (−∞, 2) ∪ (2, ∞).
Check An input such as 1.9 remains allowed, as does 2.1. The omission is one number, not a whole neighborhood.
Rung 5Notation ladder 5: overlap and a single-value set
Simplify [0, 4] ∪ [2, 6]. Also write {8} in interval notation.
The overlapping intervals cover every position from 0 through 6.Both limits are 8 and included, so this is the one-member set {8}.
We need a different description with exactly the same allowed members.State what is being found before choosing the calculation.
The first interval includes every real number from 0 through 4, and the second from 2 through 6.Both use brackets, so all listed endpoints belong.
The pieces overlap from 2 through 4, leaving no gap between 0 and 6.Union includes elements in either piece and does not duplicate shared elements.
Write the union as [0, 6].Its smallest member is 0 and largest is 6, and every number between is included.
Write {8} as [8, 8].The only number between equal included endpoints is that endpoint itself.
Answer
Union: [0, 6].
Single-value interval: [8, 8].
Check 5 comes from the second interval and 1 from the first. In [8, 8], 8 satisfies both limits, while 7.9 and 8.1 do not.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Infinity belongs, so write [5, ∞].
Infinity is not a real number or a reachable endpoint. The left endpoint also depends on whether equality is allowed.
✓ Instead: For x ≥ 5 write [5, ∞); for x > 5 write (5, ∞).
✗ Not this: Union applies only when sets are separate.
Union keeps all members in either set, whether the sets overlap or not.
✓ Instead: [0, 4] ∪ [2, 6] = [0, 6].
Tips and tricks
Translate one endpoint at a time: equality means included, and strict means excluded.
Write or next to ∪ as a memory cue.
Test one boundary and one gap value to check that a notation translation has kept the same set.
Trap. Writing [1, ∞) for [1, 3] ∪ (5, ∞). That would fill the missing region after 3 through 5. The union symbol joins sets without filling gaps.
