Number Lines, Intervals, and Allowed Values, foundations for calculus lesson

A ride at a fair admits riders at least 120 centimeters tall but under 190. That sentence describes a stretch of the number line with one end included and the other end left out. Calculus talks about such stretches all the time, when it asks where a function is defined, where it increases or what happens just to one side of a point. Every bracket and every endpoint is a yes or no decision about membership, and this lesson is about making each one on purpose.

Key terms

Real number line

Definition The real number line is a line on which each point corresponds to exactly one real number and each real number to exactly one point. A chosen point stands for $0$ , a chosen unit length fixes where $1$ is, and numbers increase to the right. For real numbers $a$ and $b$ , the statement $a < b$ means that $a$ lies to the left of $b$ . Unless a problem says otherwise, every variable in this course ranges over the real numbers.

In plain words Every real number has its own spot on the line, including fractions and numbers like $\sqrt{2}$ and $\pi$ , with no gaps between them.

Example The numbers $-2$ , $-\tfrac{1}{2}$ , $\sqrt{2} \approx 1.414$ and $\pi \approx 3.142$ sit on the line in that order from left to right. Note that $-2 < -\tfrac{1}{2}$ , because $-2$ is farther left, even though $2$ is larger than $\tfrac{1}{2}$ .

Inequality

Definition An inequality compares two real quantities with one of the symbols $<$ , $\le$ , $>$ or $\ge$ . A strict inequality ( $<$ or $>$ ) excludes equality, so $x < 3$ is false when $x = 3$ . A weak inequality ( $\le$ or $\ge$ ) includes equality, so $x \le 3$ is true when $x = 3$ . A double inequality $a < x \le b$ means that both $a < x$ and $x \le b$ hold. When you multiply or divide both sides of an inequality by a negative number, the direction of the inequality reverses.

In plain words The line under the symbol says the boundary number itself is allowed. No line means the boundary is left out.

Example The ride condition reads $120 \le x < 190$ , where $x$ is height in centimeters. A rider of exactly $120$ centimeters gets on, and a rider of exactly $190$ does not. Solving $-2x > 6$ means dividing by $-2$ , which reverses the sign and gives $x < -3$ . A check with $x = -4$ gives $8 > 6$ , which is true.

Interval

Definition An interval is a set of real numbers that contains every number lying between any two of its members. For real numbers $a < b$ the bounded intervals are written as follows.

Interval	Meaning
$(a, b)$	$a < x < b$
$[a, b]$	$a \le x \le b$
$[a, b)$	$a \le x < b$
$(a, b]$	$a < x \le b$

The numbers $a$ and $b$ are the endpoints. The notation $(a, b)$ for an interval is the same as for a point in the plane, and the context tells you which is meant.

The definition also admits sets that have no two different members, because for them there is nothing to check. A single point $\{a\} = [a, a]$ and the empty set both count as intervals in this sense and are called degenerate. They rarely matter, and when calculus speaks of a function on an interval it almost always means one with more than one point, and such an interval contains infinitely many numbers. The unbounded intervals in the section on infinity below also satisfy the definition.

In plain words An interval is one unbroken piece of the number line, with no gaps inside it. Interval notation names the two ends and uses the bracket shape to say whether each end belongs.

Example The condition $-1 < x \le 4$ is the interval $(-1, 4]$ . It contains $0$ , $3.9$ and $4$ , but not $-1$ and not $4.01$ . The set of allowed values $(-\infty, 3) \cup (3, \infty)$ found later on this page is not an interval, since $2$ and $4$ belong to it but $3$ , which lies between them, does not.

Open and closed endpoints

Definition An endpoint is closed when it belongs to the set. It is written with a square bracket and drawn on a number line as a filled dot. An endpoint is open when it does not belong to the set. It is written with a round parenthesis and drawn as a hollow dot. A strict inequality produces an open endpoint and a weak inequality produces a closed one.

In plain words Square bracket and filled dot mean the number is in. Round parenthesis and hollow dot mean it is out.

Example The ride interval $[120, 190)$ is drawn with a filled dot at $120$ , a hollow dot at $190$ and the segment between them shaded. Each of the three pieces of the picture answers one question. Is $120$ in? Yes. Is $190$ in? No. Is everything in between in? Yes.

Infinity in interval notation

Definition The symbols $\infty$ and $-\infty$ are not real numbers. In interval notation they record that the set has no upper or no lower bound. Neither symbol is a member of any set of real numbers, so each always gets a round parenthesis. For a real number $a$ the unbounded intervals are

\begin{aligned} (a, \infty) &= \{x \mid x > a\}, \\[0.4em] [a, \infty) &= \{x \mid x \ge a\}, \\[0.4em] (-\infty, a) &= \{x \mid x < a\}, \\[0.4em] (-\infty, a] &= \{x \mid x \le a\}, \end{aligned}

and the whole line is $(-\infty, \infty)$ .

In plain words Infinity means the shading keeps going forever in that direction. It is a direction, not a place you can stand on, so it is never bracketed.

Example The condition $x \ge -3$ is $[-3, \infty)$ . On a number line it is a filled dot at $-3$ with an arrow running to the right. Writing $[-3, \infty]$ is incorrect, because it claims that $\infty$ is a member of the set.

Union

Definition The union of two sets $A$ and $B$ is the set

A \cup B = \{x \mid x \in A \text{ or } x \in B\},

where “or” is inclusive, so a number belongs to $A \cup B$ when it belongs to at least one of the two sets. Unions describe conditions made of separate pieces, which a single interval cannot express.

In plain words A number is in the union if it passes at least one of the tests.

Example The condition ” $x < -2$ or $x \ge 5$ ” is

(-\infty, -2) \cup [5, \infty).

The number $-3$ is in it, so is $5$ , and $0$ is not. The number $-2$ is not in it either, since it fails both tests. Writing the condition as $5 \le x < -2$ would be wrong, because no number is both at least $5$ and less than $-2$ .

Allowed values

Definition The allowed values of a formula in one real variable are the real inputs for which every step of the calculation is defined in the real numbers. The two most common restrictions at this stage are that a denominator may not be $0$ and that the expression under an even root, such as a square root, must be greater than or equal to $0$ . A real situation may restrict the inputs further, for example to times $t \ge 0$ .

In plain words Allowed values are the inputs the formula can actually handle. Find what could go wrong, and remove those inputs.

Example The formula $\sqrt{x - 2}$ needs $x - 2 \ge 0$ , so its allowed values are $[2, \infty)$ . The formula $\frac{1}{x - 3}$ needs $x \ne 3$ , so its allowed values are

(-\infty, 3) \cup (3, \infty).

The formula

\frac{\sqrt{x - 2}}{x - 3}

needs both conditions at once, which gives $[2, 3) \cup (3, \infty)$ . The number $2$ is allowed, since it gives $\frac{0}{-1} = 0$ .

Check yourself

Write $-5 \le x < 1$ in interval notation and describe its number line picture.

Show the answer

The interval is $[-5, 1)$ . Draw a filled dot at $-5$ , a hollow dot at $1$ and shade the segment between them.

Write ” $x \le 0$ or $x > 3$ ” in interval notation. Is $3$ in the set? Is $0$ ?

Show the answer

The set is $(-\infty, 0] \cup (3, \infty)$ . The number $3$ is not in it, since $3 \le 0$ and $3 > 3$ are both false. The number $0$ is in it, since $0 \le 0$ is true.

Find the allowed values of

\frac{\sqrt{x + 1}}{x - 4}.

Show the answer

The root needs $x + 1 \ge 0$ , so $x \ge -1$ , and the denominator needs $x \ne 4$ . The allowed values are $[-1, 4) \cup (4, \infty)$ .

Number Lines, Intervals, and Allowed Values

Theory, part 3

Practice, part 4

Key terms

Real number line

Inequality

Interval

Open and closed endpoints

Infinity in interval notation

Union

Allowed values

Check yourself

How this lesson connects

Builds on

Used later in

Where this leads in Calculus