Intervals on the real line

The solu­tions of an equa­tion such as $2x + 1 = 7$ are a few sep­a­rate num­bers. The solu­tions of an inequal­ity such as $2x + 1 < 7$ fill a whole stretch of the num­ber line. Inter­vals are the short nota­tion for such stretches. They are used to write the answers of inequal­i­ties, and in the next course to describe where a func­tion is defined. This les­son intro­duces every kind of inter­val and shows how to move between inter­val nota­tion, inequal­i­ties, set-builder form and pic­tures on the num­ber line.

The real line#

Every real num­ber cor­re­sponds to exactly one point on a straight line, and every point to exactly one real num­ber. Num­bers increase from left to right. For two real num­bers $a$ and $b$ with $a < b$, the num­bers between them form an unbro­ken stretch of the line. The ques­tion for each inter­val is sim­ply: are the end points $a$ and $b$ included or not?

On a num­ber line, an end point that is included is drawn as a filled dot, and one that is excluded as an open (hol­low) cir­cle. The stretch between them is shaded or drawn thick.

The four bounded inter­vals#

Let $a < b$.

A round bracket means the end point is left out; a square bracket means it is put in. The open-closed and closed-open inter­vals are also called half-open inter­vals.

For exam­ple, $(2, 5)$ con­tains 2.1, 3, 4.99 and every other num­ber strictly between 2 and 5, but not 2 or 5 them­selves. $[2, 5]$ con­tains 2 and 5 as well. On the num­ber line, $(2, 5]$ is drawn with an open cir­cle at 2, a filled dot at 5, and the stretch between them shaded.

The length of each of these inter­vals is $b - a$. Includ­ing or exclud­ing the end points does not change the length.

Be care­ful: the inter­val $(2, 5)$ is not the same as the set $\{3, 4\}$. An inter­val con­tains all the real num­bers in the stretch, includ­ing frac­tions and irra­tional num­bers such as $\sqrt{5}$ and $\pi$, not just the inte­gers.

Infi­nite inter­vals#

The sym­bol $\infty$ (infin­ity) is used to show that a stretch goes on with­out end to the right, and $-\infty$ that it goes on with­out end to the left. Infin­ity is not a real num­ber, so it is never included: the bracket next to $\infty$ or $-\infty$ is always round.

For exam­ple, the set of pos­i­tive real num­bers is $(0, \infty)$, and the set of non-neg­a­tive real num­bers is $[0, \infty)$.

Exam­ple 1#

Write each inequal­ity as an inter­val: (a) $-3 \le x < 4$ (b) $x > 7$ (c) $x \le -2$ (d) $0 < x \le 1$.

(a) $[-3, 4)$ (b) $(7, \infty)$ (c) $(-\infty, -2]$ (d) $(0, 1]$

Exam­ple 2#

Write each inter­val as an inequal­ity and describe its pic­ture: (a) $(-1, 6]$ (b) $[5, \infty)$ (c) $(-\infty, 0)$.

(a) $-1 < x \le 6$: open cir­cle at $-1$, filled dot at 6, shaded between.

(b) $x \ge 5$: filled dot at 5, shaded to the right with­out end.

(c) $x < 0$: open cir­cle at 0, shaded to the left with­out end. This is the set of neg­a­tive real num­bers.

Exam­ple 3#

Write $\{x \in R : -4 < x \le 0\}$ in inter­val nota­tion, and find the length.

It is $(-4, 0]$, with length $0 - (-4) = 4$.

Exam­ple 4#

Which of the num­bers $-2$, $0$, $\displaystyle \frac{1}{2}$, $1$, $\sqrt{2}$ and $3$ belong to the inter­val $(0, \sqrt{2}]$?

The inter­val con­tains the num­bers greater than 0 and at most $\sqrt{2}$ (about 1.414). So $\displaystyle \frac{1}{2}$, $1$ and $\sqrt{2}$ belong to it. The num­bers $-2$ and $0$ do not (0 is excluded), and nei­ther does 3.

Exam­ple 5#

Write in inter­val nota­tion the set of real num­bers whose dis­tance from 3 is less than 2.

These are the num­bers between $3 - 2 = 1$ and $3 + 2 = 5$, not includ­ing the ends. So the set is $(1, 5)$.

Inter­vals and sets of inte­gers#

Some­times a prob­lem asks only for the inte­gers in an inter­val.

Exam­ple 6#

List the inte­gers in $[-2, 3)$, and find how many inte­gers lie in $(-2.5, 4.1)$.

The inte­gers $x$ with $-2 \le x < 3$ are $-2, -1, 0, 1, 2$.

The inte­gers in $(-2.5, 4.1)$ are $-2, -1, 0, 1, 2, 3, 4$: seven of them.

Prac­tice#

1. Write as inter­vals: (a) $1 < x < 9$ (b) $-5 \le x \le -1$ (c) $2 \le x < 2.5$ (d) $x \ge -3$ (e) $x < 10$
2. Write as inequal­i­ties: (a) $[0, 7)$ (b) $(-\infty, 4]$ (c) $(-3, \infty)$ (d) $[-1, 1]$
3. Describe the pic­ture of $[-2, 3)$ on the num­ber line.
4. Write $\{x \in R : x > -1\}$ and $\{x \in R : -6 \le x < -2\}$ in inter­val nota­tion.
5. Find the length of $(-3.5, 2]$.
6. Which of $-1$, $0$, $2$, $2.5$, $4$ belong to $[0, 2.5)$?
7. List the inte­gers in $(-3, 2]$.
8. How many inte­gers are in $[-4, 4]$? How many in $(-4, 4)$?
9. Write in inter­val nota­tion the set of real num­bers whose dis­tance from $-1$ is at most 3.
10. Is $\{x \in R : 4 < x < 4\}$ an inter­val? What set is it?

Answers#