Intervals, Endpoints, and Rays

The Big Idea

An interval is a connected portion of an ordered number line. If it contains two real numbers, it contains every real number between them. Its endpoints mark where the interval begins or ends, while a ray continues without bound in one direction.

The same set can be described in several useful ways:

\[ -2<x\le5 \qquad\Longleftrightarrow\qquad x\in(-2,5] \qquad\Longleftrightarrow\qquad \{x\in\mathbb R:-2<x\le5\}. \]

These are not three different answers. They are three languages for the same mathematical object.

What Is an Interval?

A subset \(I\) of the real numbers is an interval when it has no gaps between any two of its members. More precisely, if \(a,b\in I\), \(a<b\), and \(a<c<b\), then \(c\in I\).

\[ a,b\in I\text{ and }a<c<b \quad\Longrightarrow\quad c\in I. \]

For example, \([1,4]\) is an interval because it includes every real number from \(1\) through \(4\). The set \([1,2]\cup[3,4]\) is not one interval because it has a gap between \(2\) and \(3\); it is a union of two intervals.

The “no gaps” idea refers to the real-number line. When the domain is restricted to integers or another discrete set, notation must be interpreted relative to that domain.

Endpoints Are Boundary Values

An endpoint is a finite boundary value at which an interval starts or stops. For the interval \([-3,7)\), the endpoints are \(-3\) and \(7\).

The notation also tells us whether each boundary value belongs to the set:

• A square bracket includes the adjacent endpoint.
• A parenthesis excludes the adjacent endpoint.

\[ -3\in[-3,7), \qquad 7\notin[-3,7). \]

Read the symbols locally. The left mark tells you about the left endpoint, and the right mark tells you about the right endpoint.

The Four Bounded Interval Types

Let \(a<b\). Each endpoint may be included or excluded independently, giving four standard bounded intervals.

The two mixed forms are often called half-open or half-closed. Neither name tells you which endpoint is included, so inspect the actual symbols.

Brackets Mean “Included”

A bracket corresponds to a non-strict inequality:

\[ x\ge a \quad\Longleftrightarrow\quad x\in[a,\infty), \]

because \(x=a\) is permitted. Likewise,

\[ x\le b \quad\Longleftrightarrow\quad x\in(-\infty,b]. \]

On a number-line graph, included endpoints are normally shown with filled dots. A filled dot and a bracket communicate the same boundary decision.

Parentheses Mean “Excluded”

A parenthesis corresponds to a strict inequality:

\[ x>a \quad\Longleftrightarrow\quad x\in(a,\infty), \]

and

\[ x<b \quad\Longleftrightarrow\quad x\in(-\infty,b). \]

On a graph, an excluded endpoint is normally shown with an open circle. The circle marks the boundary location without adding that value to the set.

Reading a Number-Line Graph

A number-line graph combines boundary marks with a highlighted region. Read it in two stages: identify the boundary values, then decide whether each boundary is included.

Do not infer inclusion from the direction of shading. Inclusion is determined only by the boundary mark or the inequality symbol.

Rays and Half-Lines

A geometric ray starts at an endpoint and continues forever in one direction. On a number line, a rightward ray beginning at \(a\) represents all values at or beyond \(a\):

\[ [a,\infty)=\{x\in\mathbb R:x\ge a\}. \]

A leftward ray beginning at \(b\) represents

\[ (-\infty,b]=\{x\in\mathbb R:x\le b\}. \]

In elementary algebra, the sets \((a,\infty)\) and \((-\infty,b)\) are also informally called rays, although they omit the finite boundary. The term half-line is useful when we want to discuss both included and excluded boundary cases without confusion.

Infinity Is Not an Endpoint

The symbols \(\infty\) and \(-\infty\) indicate that the set continues without bound. They are not real numbers and cannot be included as members of a real interval.

Therefore a parenthesis is always used beside an infinity symbol:

\[ (4,\infty), \qquad (-\infty,9], \qquad (-\infty,\infty). \]

Writing \([4,\infty]\) would incorrectly suggest that infinity is a last real point contained in the interval. There is no greatest real number at the far end of \([4,\infty)\).

The Four Common Infinite Intervals

The Entire Real Line

Every real number is represented by

\[ \mathbb R=(-\infty,\infty). \]

This interval has no finite endpoints. Both infinity symbols receive parentheses because neither is a real member of the set.

An inequality such as \(x^2\ge0\) is true for every real \(x\), so its real solution set is \((-\infty,\infty)\).

From Inequalities to Intervals

To translate a compound inequality, place the smaller boundary on the left and the larger boundary on the right. Then convert each inequality symbol into the matching endpoint mark.

From Intervals to Inequalities

Read interval notation from left to right. Parentheses produce strict inequalities, while brackets produce inclusive inequalities.

\[ x\in[-1,6) \quad\Longleftrightarrow\quad -1\le x<6. \]

For an infinite interval, keep the variable beside the finite boundary:

\[ x\in(-\infty,3] \quad\Longleftrightarrow\quad x\le3. \]

A useful check is to choose one interior value and test it. The value \(0\) belongs to \([-1,6)\), and it satisfies \(-1\le0<6\).

Set-Builder Notation Makes the Domain Explicit

Interval notation normally assumes a real-number domain. Set-builder notation can state that domain directly:

\[ [2,5) = \{x\in\mathbb R:2\le x<5\}. \]

The colon is read “such that.” A vertical bar is also common:

\[ \{x\in\mathbb R\mid 2\le x<5\}. \]

Stating the domain prevents ambiguity when the same inequality is being interpreted over real numbers, integers, rational numbers, dates, or measured quantities.

“And” Selects an Overlap

A chained inequality such as \(2<x\le8\) means that both conditions must hold:

\[ x>2 \quad\text{and}\quad x\le8. \]

In set language, “and” corresponds to intersection:

\[ (2,\infty)\cap(-\infty,8]=(2,8]. \]

Only values in both component sets survive. The overlap is one bounded interval.

“Or” Combines Regions

The condition \(x<-1\) or \(x\ge4\) describes two separated rays. It cannot be written as one interval because the values between \(-1\) and \(4\) are missing.

\[ (-\infty,-1)\cup[4,\infty). \]

The union symbol \(\cup\) means that a value may belong to either set or to both. It joins the listed pieces without filling the gap between them.

This structure appears naturally when an absolute-value inequality describes values outside a central interval:

\[ |x|>3 \quad\Longleftrightarrow\quad x\in(-\infty,-3)\cup(3,\infty). \]

Finding Intersections

To find an intersection, identify the part common to both sets. Begin with the more restrictive left boundary and end with the more restrictive right boundary.

\[ [-4,6)\cap[1,9]=[1,6). \]

At \(1\), both sets include the value, so the result uses a bracket. At \(6\), the first set excludes the value, so the result uses a parenthesis.

Some intervals do not overlap:

\[ (-2,0)\cap[0,5]=\varnothing. \]

Although the pieces meet visually at \(0\), the first interval does not contain \(0\), so there is no shared member.

Combining Overlapping or Touching Intervals

Overlapping intervals combine into one larger interval:

\[ [1,5]\cup(3,8)=[1,8). \]

Touching intervals combine into one interval when their shared boundary is supplied by at least one piece:

\[ [1,3]\cup(3,7]=[1,7]. \]

But if both pieces exclude the meeting point, a hole remains:

\[ [1,3)\cup(3,7] \]

cannot be collapsed to \([1,7]\), because \(3\) is missing.

Empty Intervals and Single Points

When the two written boundaries are equal, endpoint inclusion becomes decisive.

The symbol \(\varnothing\) denotes the empty set. It is a set with no members; it is not the same object as the number zero.

What If the Boundaries Are Reversed?

Standard interval notation lists the lesser boundary first. If \(a>b\), the condition

\[ a\le x\le b \]

cannot be satisfied, so the described set is empty. An expression such as \([8,2]\) should not be silently reinterpreted as \([2,8]\); it is malformed under some elementary conventions and treated as empty under the set-defined convention.

First determine the mathematical condition. Then write the lower boundary on the left and the upper boundary on the right.

The Domain Changes the Members

Over the real numbers,

\[ [2,5]=\{x\in\mathbb R:2\le x\le5\} \]

contains infinitely many values, including \(2.1\), \(\sqrt8\), and \(4.999\ldots\). Over the integers, the same boundary condition selects only

\[ \{x\in\mathbb Z:2\le x\le5\}=\{2,3,4,5\}. \]

Interval notation is designed primarily for subsets of \(\mathbb R\). When the intended domain is discrete, state it explicitly or list the values when practical.

The Word “Between” Needs Clarification

In ordinary language, “between \(a\) and \(b\)” may or may not include the boundary values. Mathematics should not leave that decision hidden.

• Strictly between normally means \(a<x<b\), or \((a,b)\).
• Between, inclusive means \(a\le x\le b\), or \([a,b]\).
• From \(a\) through \(b\) often suggests inclusion, but the context should confirm it.
• More than \(a\) but no more than \(b\) means \(a<x\le b\), or \((a,b]\).

When writing a problem, specify the inequalities or use words such as inclusive, exclusive, at least, or less than.

Boundary Language in Applications

Context can refine these translations. For example, legal deadlines, machine tolerances, and ticket eligibility rules may define boundaries differently, so consult the actual rule before modeling it.

Length and Midpoint of a Bounded Interval

For finite real boundaries \(a\le b\), the span from \(a\) to \(b\) has length

\[ L=b-a. \]

Its midpoint is

\[ m=\frac{a+b}{2}. \]

The intervals \((a,b)\), \([a,b]\), \([a,b)\), and \((a,b]\) have the same geometric span \(b-a\). Including or excluding individual endpoints does not change real interval length, even though it changes set membership.

For \([-3,9)\), the span is \(9-(-3)=12\), and the midpoint of the two boundary values is \(3\).

Center-Radius Form

A bounded interval can be described by its midpoint \(m\) and half-length \(r\):

\[ m=\frac{a+b}{2}, \qquad r=\frac{b-a}{2}. \]

The closed interval \([a,b]\) becomes

\[ [a,b]=[m-r,m+r] =\{x\in\mathbb R:|x-m|\le r\}. \]

The open interval \((a,b)\) similarly becomes \(|x-m|<r\). This connection turns interval membership into a statement about distance from a center.

Counting Integers Between Endpoints

Geometric length and discrete counting are different tasks. If \(a\) and \(b\) are integers with \(a\le b\), the closed integer range contains

\[ b-a+1 \]

integers. Thus \(\{4,5,6,7,8\}\) contains \(8-4+1=5\) integers, although the distance from \(4\) to \(8\) is \(4\).

These formulas assume integral endpoints and the integer domain. For nonintegral boundaries, use ceiling and floor or list the eligible integers.

Time Windows and Scheduling

Half-open intervals are especially useful for schedules. Suppose one shift runs from 8:00 a.m. until 12:00 p.m., and the next begins at 12:00 p.m. A clean model is

\[ [8,12) \qquad\text{and}\qquad [12,16). \]

Noon belongs to the second shift but not the first, so the assignments neither overlap nor leave a gap. Computer programs frequently use half-open index and time ranges for the same reason.

Measurement and Tolerance Intervals

A component designed to be \(20\) millimeters long with an allowed deviation of at most \(0.05\) millimeter must satisfy

\[ |x-20|\le0.05. \]

In interval form,

\[ x\in[19.95,20.05]. \]

The endpoints are included because the deviation may be exactly \(0.05\). If the specification said “less than \(0.05\),” the acceptable interval would be open.

Intervals as Domains and Ranges

Intervals often describe the inputs or outputs of a function. For

\[ f(x)=\sqrt{x-3}, \]

real outputs require \(x-3\ge0\). Therefore the real domain is

\[ [3,\infty). \]

The function can produce \(0\), and all its outputs are nonnegative, so its range is also \([0,\infty)\).

A formula may require several restrictions. The final domain is the intersection of all permitted input sets.

Intervals Can Partition a Number Line

A partition divides a set into nonoverlapping pieces whose union is the original set. For example,

\[ (-\infty,0)\cup\{0\}\cup(0,\infty)=\mathbb R. \]

The three pieces are pairwise disjoint: negative numbers, zero, and positive numbers. Piecewise-defined functions use interval partitions to specify which rule applies to each input.

Boundary decisions matter. If two pieces both contain a shared endpoint, the rules overlap there. If neither contains it, the definition has a gap.

A First Look at Open and Closed Sets

In elementary interval language, “open” means both finite endpoints are excluded, and “closed” means both are included. Later, topology gives these words broader definitions based on neighborhoods and limit points.

• \((a,b)\) is open in \(\mathbb R\).
• \([a,b]\) is closed in \(\mathbb R\).
• \((a,b]\) and \([a,b)\) are neither open nor closed in \(\mathbb R\).
• \((a,\infty)\) is open, while \([a,\infty)\) is closed in \(\mathbb R\).
• \(\mathbb R\) and \(\varnothing\) are both open and closed in \(\mathbb R\).

This later meaning depends on the surrounding space. The basic bracket rule remains the right tool for reading whether a listed finite boundary belongs to an interval.

Intervals Can Represent Uncertainty

Sometimes a quantity is not known exactly but is known to lie within a range. If \(x\in[2,3]\) and \(y\in[5,7]\), then every possible sum lies in

\[ x+y\in[2+5,3+7]=[7,10]. \]

This is a first example of interval arithmetic. It tracks guaranteed bounds rather than pretending an uncertain quantity has one exact value. Multiplication, division, and dependent quantities require additional care, which later lessons can develop.

Test Membership Directly

To decide whether a value belongs to an interval, substitute it into the corresponding inequalities.

For \(I=(-2,5]\):

• \(-2\notin I\) because the left endpoint is excluded.
• \(0\in I\) because \(-2<0\le5\).
• \(5\in I\) because the right endpoint is included.
• \(6\notin I\) because \(6>5\).

Endpoint testing is especially effective for catching a bracket or inequality-sign error.

A Reliable Interval Routine

1. Identify the domain. Determine whether the variable represents real numbers, integers, time, measurements, or another set.
2. Locate the finite boundaries. Put the smaller boundary on the left and the larger boundary on the right.
3. Decide inclusion separately. Use brackets for included finite boundaries and parentheses for excluded boundaries.
4. Use parentheses at infinity. Infinity indicates unbounded continuation, not a member.
5. Choose intersection for “and.” Keep only the overlap.
6. Choose union for “or.” Include all permitted pieces without filling gaps.
7. Test the endpoints and an interior point. Confirm that the notation matches the original condition.
8. Check units and context. Make sure the boundary rule makes practical sense.

Common Confusions

Working Summary

An interval is a gap-free portion of the real number line. Finite endpoints are included with brackets and excluded with parentheses. A ray or half-line continues without bound in one direction, and infinity always receives a parenthesis because it is not a real endpoint.

Inequality notation, interval notation, set-builder notation, and number-line graphs can represent the same set. “And” corresponds to intersection; “or” corresponds to union. The domain must be stated when the intended values are not all real numbers.

Boundary choices matter in scheduling, tolerances, eligibility rules, function domains, piecewise definitions, and computation. Translate carefully, test endpoints directly, and preserve gaps rather than allowing compact notation to hide them.

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