The Central Idea
Suppose the solution to an inequality is
\[ x<5. \]This same set can be written in several ways.
Inequality notation:
\[ x<5. \]Interval notation:
\[ (-\infty,5). \]Set-builder notation:
\[ \{x\in\mathbb{R}\mid x<5\}. \]Graphically: every point to the left of \(5\), with \(5\) excluded.
These are not different answers. They are different representations of the same set.
Why Multiple Notations Matter
Different representations emphasize different information.
- Inequality notation shows the comparison directly.
- Interval notation makes connected regions and endpoints compact.
- Set-builder notation states the rule defining membership.
- Graphical notation makes location, inclusion, gaps, and unions visually immediate.
Learning to translate among these forms is part of understanding the set itself.
Interval Notation
Interval notation describes connected portions of the real number line using endpoints and parentheses or brackets.
For example,
\[ 2The parentheses indicate that the endpoints are excluded.
Parentheses Mean Excluded Endpoints
The interval
\[ (2,7) \]contains every real number strictly between \(2\) and \(7\), but not the endpoints themselves.
So it is equivalent to
\[ 2Brackets Mean Included Endpoints
The interval
\[ [2,7] \]contains every real number between \(2\) and \(7\), including both endpoints.
It is equivalent to
\[ 2\le x\le7. \]Mixed Endpoints
The interval
\[ (2,7] \]excludes \(2\) but includes \(7\).
It is equivalent to
\[ 2means
\[ 2\le x<7. \]Unbounded Intervals
The inequality
\[ x>4 \]extends indefinitely to the right.
Its interval notation is
\[ (4,\infty). \]The inequality
\[ x\le4 \]extends indefinitely to the left:
\[ (-\infty,4]. \]Infinity Always Uses Parentheses
The symbols
\[ \infty \]and
\[ -\infty \]are not ordinary real-number endpoints.
We do not “reach” infinity and include it.
Therefore, interval notation always uses parentheses with infinity:
\[ (-\infty,3], \] \[ (5,\infty), \]never
\[ [-\infty,3] \]or
\[ [5,\infty]. \]The Entire Real Line
The set of all real numbers is written
\[ \mathbb{R}. \]In interval notation, the same set is
\[ (-\infty,\infty). \]Both notations describe the complete real number line.
The Empty Set
If no real values satisfy a condition, the solution set is the empty set.
It is written
\[ \varnothing. \]For example,
\[ x<2 \quad\text{and}\quad x>5 \]has no real solution, so
\[ \boxed{\varnothing}. \]A Single Point Is Not an Interval
If the solution is only
\[ x=4, \]the solution set is
\[ \{4\}. \]This is a singleton set.
Although some extended conventions permit degenerate interval notation such as \([4,4]\), the set notation
\[ \{4\} \]is usually clearer when the solution is one isolated point.
Set-Builder Notation
Set-builder notation describes a set by stating the condition its elements must satisfy.
For example,
\[ \{x\in\mathbb{R}\mid x>3\} \]means:
“the set of all real numbers \(x\) such that \(x>3\).”
The Vertical Bar in Set-Builder Notation
The symbol
\[ \mid \]in
\[ \{x\in\mathbb{R}\mid x>3\} \]means “such that.”
A colon is also sometimes used:
\[ \{x\in\mathbb{R}:x>3\}. \]Both forms describe the same set.
Set Membership
The symbol
\[ \in \]means “is an element of.”
Thus,
\[ x\in\mathbb{R} \]means that \(x\) is a real number.
Set-builder notation often begins by stating the universe in which the variable is allowed to vary.
Set-Builder Example
The interval
\[ [-2,5) \]can be written as
\[ \{x\in\mathbb{R}\mid -2\le x<5\}. \]The interval notation emphasizes endpoints.
The set-builder notation emphasizes the defining condition.
Set-Builder Notation Is More General Than Interval Notation
Interval notation is especially useful for connected regions of the real number line.
Set-builder notation can describe much more general sets.
For example,
\[ \{n\in\mathbb{Z}\mid n\text{ is even}\} \]describes all even integers.
This set cannot be represented naturally by one ordinary interval because its elements are discrete rather than continuous.
Specify the Number System When It Matters
Compare
\[ \{x\in\mathbb{R}\mid 0The second contains only
\[ \{1,2,3,4\}. \]The condition may look similar, but the underlying domain changes the set completely.
Graphical Number-Line Notation
A number-line graph shows the solution set spatially.
For a boundary point:
- an open point means the boundary is excluded;
- a closed point means the boundary is included.
The shaded or highlighted region shows all allowed values.
Graphing \(x<3\)
For
\[ x<3, \]place an open boundary at \(3\) and extend the solution region to the left.
The equivalent interval is
\[ (-\infty,3). \]The equivalent set-builder form is
\[ \{x\in\mathbb{R}\mid x<3\}. \]Graphing \(x\ge3\)
For
\[ x\ge3, \]place a closed boundary at \(3\) and extend the solution region to the right.
Interval notation:
\[ [3,\infty). \]Set-builder notation:
\[ \{x\in\mathbb{R}\mid x\ge3\}. \]Graphing a Bounded Interval
Consider
\[ -2- an open boundary at \(-2\);
- a closed boundary at \(5\);
- the region between them included.
Interval notation:
\[ (-2,5]. \]Unions of Intervals
Some solution sets contain separated regions.
For example,
\[ x<-3 \quad\text{or}\quad x>2 \]is written
\[ (-\infty,-3)\cup(2,\infty). \]The union symbol
\[ \cup \]means that values from either interval belong to the set.
Union in Set-Builder Notation
The set
\[ (-\infty,-3)\cup(2,\infty) \]can be written as
\[ \{x\in\mathbb{R}\mid x<-3\text{ or }x>2\}. \]The logical word or corresponds naturally to set union.
Intersections
Suppose
\[ x>1 \]and
\[ x\le6. \]The values satisfying both conditions form the intersection:
\[ (1,\infty)\cap(-\infty,6]. \]This simplifies to
\[ (1,6]. \]The symbol
\[ \cap \]means intersection.
Intersection in Set-Builder Notation
The interval
\[ (1,6] \]is equivalent to
\[ \{x\in\mathbb{R}\mid x>1\text{ and }x\le6\}. \]The logical word and corresponds naturally to intersection.
Excluded Points Inside an Interval
Suppose
\[ x>-4 \]but
\[ x\ne2. \]The solution set is
\[ (-4,2)\cup(2,\infty). \]The missing point creates a hole that splits one larger interval into two pieces.
Set Difference
The same set can be written as
\[ (-4,\infty)\setminus\{2\}. \]The symbol
\[ \setminus \]means “remove from the set.”
This notation is useful when only a few isolated values are excluded from an otherwise simple region.
Domain Restrictions in Set Notation
For the rational expression
\[ \frac{1}{x-5}, \]the domain is
\[ \mathbb{R}\setminus\{5\}. \]In interval notation:
\[ (-\infty,5)\cup(5,\infty). \]In set-builder notation:
\[ \{x\in\mathbb{R}\mid x\ne5\}. \]Multiple Excluded Points
For
\[ \frac{1}{(x-2)(x+4)}, \]the domain excludes
\[ x=2 \]and
\[ x=-4. \]Set notation:
\[ \mathbb{R}\setminus\{-4,2\}. \]Interval notation:
\[ (-\infty,-4)\cup(-4,2)\cup(2,\infty). \]Translate from Inequality to Interval
Example:
\[ -3\le x<8. \]The left endpoint is included, so use a bracket.
The right endpoint is excluded, so use a parenthesis.
Therefore,
\[ [-3,8). \]Translate from Interval to Inequality
Example:
\[ (-5,2]. \]The interval begins just above \(-5\) and includes \(2\).
So
\[ -5Translate from Set-Builder to Interval
Consider
\[ \{x\in\mathbb{R}\mid x\ge1\}. \]This means every real number beginning at \(1\) and extending rightward.
Interval notation:
\[ [1,\infty). \]Translate from Interval to Set-Builder
Consider
\[ (-\infty,-2)\cup[4,\infty). \]Set-builder notation:
\[ \{x\in\mathbb{R}\mid x<-2\text{ or }x\ge4\}. \]Compound Inequalities and Interval Notation
An and inequality usually becomes one interval:
\[ 2An or inequality often becomes a union:
\[ x<-1 \quad\text{or}\quad x\ge5 \]becomes
\[ (-\infty,-1)\cup[5,\infty). \]Absolute-Value Inequalities
Absolute-value inequalities often translate naturally into intervals.
For example,
\[ |x-3|<4 \]means
\[ -1Set-builder notation:
\[ \{x\in\mathbb{R}\mid -1Absolute Value Outside an Interval
Consider
\[ |x-3|\ge4. \]This means
\[ x\le-1 \]or
\[ x\ge7. \]Interval notation:
\[ (-\infty,-1]\cup[7,\infty). \]Set-builder notation:
\[ \{x\in\mathbb{R}\mid x\le-1\text{ or }x\ge7\}. \]Polynomial-Inequality Solutions
Suppose sign analysis gives
\[ x\le-2 \quad\text{or}\quad x\ge5. \]The interval form is
\[ (-\infty,-2]\cup[5,\infty). \]The set-builder form is
\[ \{x\in\mathbb{R}\mid x\le-2\text{ or }x\ge5\}. \]Rational-Inequality Solutions
Suppose a rational inequality gives
\[ -1\le x<4, \qquad x\ne2. \]The interval notation is
\[ [-1,2)\cup(2,4). \]The set-builder form is
\[ \{x\in\mathbb{R}\mid -1\le x<4,\ x\ne2\}. \]Different notations can make different structural features easier to see.
Discrete Sets
Interval notation is designed for continuous stretches of the real number line.
For discrete sets, roster or set-builder notation is often better.
For example, the integers satisfying
\[ 1\le n\le5 \]are
\[ \{1,2,3,4,5\}. \]Set-builder notation:
\[ \{n\in\mathbb{Z}\mid 1\le n\le5\}. \]Roster Notation
Roster notation lists elements explicitly.
For example,
\[ \{-2,0,3,7\}. \]Roster notation is useful for small finite sets but impractical for large or infinite sets.
Set-builder notation is usually more efficient when a rule describes the elements.
Equivalent Set Descriptions
The same set might be written as
\[ \{1,2,3,4\}, \] \[ \{n\in\mathbb{Z}\mid 1\le n\le4\}, \]or verbally as:
“the integers from 1 through 4.”
Representation changes; membership does not.
Endpoint Order Matters
Intervals are normally written from smaller endpoint to larger endpoint.
Thus,
\[ (2,8) \]is standard.
Writing
\[ (8,2) \]does not mean “all numbers between 8 and 2.”
In standard real interval notation, the smaller endpoint comes first.
Why \(8<x<2\) Has No Real Solutions
The compound inequality
\[ 8No real number can satisfy both.
Therefore,
\[ \boxed{\varnothing}. \]This is not “all numbers between 2 and 8.” The direction of the inequalities matters.
Graphical Reading from Left to Right
Number-line graphs are normally read from smaller values on the left to larger values on the right.
This helps interpret intervals:
\[ [-3,4) \]starts at \(-3\), includes it, continues rightward, and stops just before \(4\).
The visual direction reinforces the order structure of the real line.
Boundary Points and Graphs
Boundary points deserve special attention because their inclusion determines whether a graph uses an open or closed point.
For
\[ x>2, \]the boundary \(2\) is excluded.
For
\[ x\ge2, \]the boundary is included.
That one symbol changes the set.
Undefined Points Are Graphically Open
If
\[ x\ne3 \]because an expression is undefined there, then the graph must leave \(3\) out of the set.
Even if nearby values on both sides are allowed, \(3\) itself remains a missing point.
This often appears as a hole in a number-line solution graph.
Comparing Representations
| Description | Inequality | Interval | Set-Builder |
|---|---|---|---|
| Less than 4 | \(x<4\) | \((-\infty,4)\) | \(\{x\in\mathbb{R}\mid x<4\}\) |
| At least 4 | \(x\ge4\) | \([4,\infty)\) | \(\{x\in\mathbb{R}\mid x\ge4\}\) |
| Between 1 and 6, both included | \(1\le x\le6\) | \([1,6]\) | \(\{x\in\mathbb{R}\mid 1\le x\le6\}\) |
| Outside \([-2,3]\) | \(x<-2\) or \(x>3\) | \((-\infty,-2)\cup(3,\infty)\) | \(\{x\in\mathbb{R}\mid x<-2\text{ or }x>3\}\) |
A Reliable Translation Workflow
- Identify the domain.
- Locate all finite boundary points.
- Determine whether each boundary is included or excluded.
- Determine which regions of the number line belong to the set.
- Use brackets for included finite endpoints and parentheses for excluded ones.
- Use union for separated regions.
- Use set-builder notation when a condition describes the set more naturally than intervals do.
- Check that all representations describe exactly the same members.
Worked Example: Three Representations
Represent the solution
\[ -4\le x<2 \]in three forms.
Inequality:
\[ -4\le x<2. \]Interval:
\[ [-4,2). \]Set-builder:
\[ \{x\in\mathbb{R}\mid -4\le x<2\}. \]Graphically, the region begins with a closed boundary at \(-4\), extends rightward, and ends with an open boundary at \(2\).
Worked Example: Union
Represent
\[ x\le-1 \quad\text{or}\quad x>5. \]Interval notation:
\[ (-\infty,-1]\cup(5,\infty). \]Set-builder notation:
\[ \{x\in\mathbb{R}\mid x\le-1\text{ or }x>5\}. \]Graphically, the solution contains a left ray ending at an included \(-1\) and a right ray beginning just beyond \(5\).
Worked Example: Excluded Point
Represent all real numbers except \(3\).
Set notation:
\[ \mathbb{R}\setminus\{3\}. \]Interval notation:
\[ (-\infty,3)\cup(3,\infty). \]Set-builder notation:
\[ \{x\in\mathbb{R}\mid x\ne3\}. \]Worked Example: Discrete Domain
Represent the integers satisfying
\[ -2Set-builder notation:
\[ \{n\in\mathbb{Z}\mid -2Common Errors
Using Brackets with Infinity
Infinity is never included as a real endpoint.
Using a Bracket for a Strict Inequality
If the boundary is excluded, use a parenthesis.
Using a Parenthesis for an Inclusive Boundary
If equality is allowed and the expression is defined there, use a bracket.
Including Undefined Values
An undefined point remains excluded even in an inclusive inequality.
Forgetting the Union Symbol
Separated intervals must be joined with
\[ \cup. \]Using Interval Notation for a Discrete Set Without Stating the Domain
Intervals normally describe real-number continua, not isolated integer values.
Reversing Endpoint Order
Write intervals from smaller value to larger value.
Confusing an Empty Set with All Real Numbers
A contradictory condition gives
\[ \varnothing, \]not
\[ \mathbb{R}. \]Practice
1. Inequality to Interval
Write in interval notation:
\[ x\ge-3. \]Check Answer
\[ [-3,\infty). \]2. Interval to Inequality
Write as an inequality:
\[ (-2,7]. \]Check Answer
\[ -23. Set-Builder to Interval
Write in interval notation:
\[ \{x\in\mathbb{R}\mid x<1\}. \]Check Answer
\[ (-\infty,1). \]4. Union
Write in interval notation:
\[ x<-4 \quad\text{or}\quad x\ge2. \]Check Answer
\[ (-\infty,-4)\cup[2,\infty). \]5. Excluded Point
Write all real numbers except \(5\) in interval notation.
Check Answer
\[ (-\infty,5)\cup(5,\infty). \]6. Set-Builder
Write
\[ [1,6) \]in set-builder notation.
Check Answer
\[ \{x\in\mathbb{R}\mid 1\le x<6\}. \]7. Empty Set
What is the solution set of
\[ 8Check Answer
\[ \varnothing. \]8. Discrete Domain
List the integer solutions to
\[ 0Key Ideas
- A single solution set can be represented with inequalities, interval notation, set-builder notation, roster notation, or graphs.
- Parentheses exclude finite endpoints; brackets include them.
- Infinity and negative infinity always use parentheses.
- Set-builder notation describes a set by specifying a domain and a membership condition.
- And corresponds to intersection; or corresponds to union.
- Separated real-number regions are written as unions of intervals.
- Undefined values must be excluded even when nearby values are included.
- Set difference is useful for removing isolated points from larger sets.
- Interval notation is best suited to continuous real-number regions.
- Set-builder or roster notation is often better for discrete domains such as the integers.
- Graphical notation makes boundaries, inclusion, gaps, and direction immediately visible.
- Equivalent representations must describe exactly the same members.
Where This Leads
Interval, set-builder, and graphical notation provide a common language for describing solution sets across equations, inequalities, functions, domains, ranges, and constraints.
These representations will continue to appear in systems of inequalities, function domains and ranges, piecewise definitions, calculus intervals, probability events, and higher-dimensional solution regions.