INTERVALS Some useful words for inequalities are: Less than “<” Greater than “>” Less than or equal to “≤ " At most “≤ " Greater than or equal to “≥” At least “≥” Not equal to “≠” In mathematics, a collection of elements is called a SET, and the symbols “ { } ” are used to enclose the elements of the set. An INTERVAL is a set of real numbers between two points, “a” and “b”, called the ENDPOINTS of the interval. We can use different ways to represent an interval: interval notation, set-builder notation and a graph on the real number line.

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The endpoints used in interval notation are always written from left to right. That is, the smallest number is written first, followed by a comma, followed by the largest number.

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Parenthesis “( “ or “)” indicate that the endpoint is excluded (not included) from the set. Parenthesis correspond to an open dot on the number line.

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Square brackets “[“ or “]” mean that the endpoint in included in the set. Square brackets correspond to a closed dot on the number line.

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Parenthesis are always used with “∞” and “−∞”.

TYPE OF INTERVAL AND GRAPH

INTERVAL NOTATION

SET-BUILDER NOTATION

OPEN INTERVAL

(𝑎 , 𝑏)

{𝑥 ∈ ℝ|𝑎 < 𝑥 < 𝑏}

[𝑎 , 𝑏]

{𝑥 ∈ ℝ|𝑎 ≤ 𝑥 ≤ 𝑏}

(𝑎 , 𝑏]

{𝑥 ∈ ℝ|𝑎 < 𝑥 ≤ 𝑏}

[𝑎 , 𝑏)

{𝑥 ∈ ℝ|𝑎 ≤ 𝑥 < 𝑏}

(𝑎 , ∞)

{𝑥 ∈ ℝ|𝑥 > 𝑎}

(−∞ , 𝑏)

{𝑥 ∈ ℝ|𝑥 < 𝑏}

[𝑎 , ∞)

{𝑥 ∈ ℝ|𝑥 ≥ 𝑎}

(−∞ , 𝑏]

{𝑥 ∈ ℝ|𝑥 ≤ 𝑏}

CLOSED INTERVAL

HALF-OPEN (OR HALF-CLOSED) INTERVALS

OPEN INFINITE INTERVALS OPEN HALF-LINE or OPEN RAY

CLOSED INFINITE INTERVALS CLOSED HALF-LINE or CLOSED RAY

EXERCISES: 1) Graph each set on the number line, and express them in interval notation: a) {𝑥 ∈ ℝ ∶ −1 < 𝑥 ≤ 5}

h) {𝑥 ∈ ℝ ∶ 𝑥 ≥ 2}

b) {𝑥 ∈ ℝ ∶ −7 < 𝑥}

i) {𝑥 ∈ ℝ ∶ 𝑥 < −5}

c) {𝑥 ∈ ℝ ∶ 𝑥 ≠ −8}

j) {𝑥 ∈ ℝ ∶ 𝑥 ≤ 4}

d) {𝑥 ∈ ℝ ∶ 𝑥 ≤ −7 𝑜𝑟 𝑥 ≥ 3}

k) {𝑥 ∈ ℝ ∶ −6 ≤ 𝑥 𝑜𝑟 𝑥 ≥ 5}

e) {𝑥 ∈ ℝ ∶ 𝑥 ≤ −8}

l) {𝑥 ∈ ℝ ∶ 𝑥 ≤ 5 𝑜𝑟 𝑥 > 6}

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f) {𝑥 ∈ ℝ ∶ − 4 ≤ 𝑥 ≤ 6}

m) {𝑥 ∈ ℝ ∶ −9 ≤ 𝑥 ≤ 0}

g) {𝑥 ∈ ℝ ∶ 𝑥 > −3}

n) {𝑥 ∈ ℝ ∶ 0 ≤ 𝑥 < 5}

2) Graph each interval and write them in set-builder notation: a) (−8 , 15)

f) [3 , 10]

k) (−∞ , −5)

b) (−2, 2)

g) [−2 , 14]

l) (3 , 12]

c) (−∞ , 2)

h) [−4, ∞)

m) (−2 , 5) ∪ (7 , ∞)

d) (−∞ , 1)

i) [−8, ∞)

n) (−∞ , −1] ∪ [7 , 10]

e) (−1 , 5]

j) (0 , 7)

o) [−8, 5) ∪[7 , 20]

3) Complete the following chart:

Interval notation

Set-builder notation

{𝑥 ∈ ℝ ∶ −7 < 𝑥 ≤ 2}

(−3 , 8]

{𝑥 ∈ ℝ ∶ 𝑥 ≠ 5}

[−10 , −4]

(−∞ , −1] ∪ [7 , 10]

Graph