This document contains exercises involving intervals of real numbers. It asks the student to represent intervals using set-builder and interval notation, graph intervals on a number line, find intersections and unions of intervals, and determine whether numbers are elements of given intervals. The exercises focus on developing skills with representing, manipulating, and reasoning about intervals of real numbers.

1. UNIT 01 September
1. Express these intervals using set-builder notation. Represent them on the number line.
a) [ 𝟐𝟐, 𝟓𝟓) =
b) �– 𝟐𝟐, +∞� =
c) �– 𝟑𝟑, 𝟖𝟖� =
d) (– ∞, – 𝟔𝟔) =
2. Write the following sets of real numbers as intervals and represent them on the
number line.
a) Real numbers between 5 and 6, both included.
b) Real numbers greater than 7.
c) Real numbers less than or equal to -5.
3. Write in set notation and represent on the number line the following intervals:
(−𝟐𝟐, 𝟏𝟏𝟏𝟏); ( 𝟐𝟐, 𝟏𝟏𝟏𝟏]; [−𝟒𝟒, −𝟏𝟏];(−∞, −𝟐𝟐); [ 𝟔𝟔, ∞)
Axel Cotón Gutiérrez Mathematics 4º ESO Exercises 4.1.3.1

2. UNIT 01 September
4. Write as intervals:
a) {𝐱𝐱 ∈ ℝ ∕ 𝐱𝐱 ≤ 𝟏𝟏𝟏𝟏}
b) {𝐱𝐱 ∈ ℝ −𝟐𝟐⁄ < 𝐱𝐱 ≤ 𝟑𝟑𝟑𝟑}
c) �𝐱𝐱 ∈ ℝ ∕ 𝐱𝐱 ≤ √𝟑𝟑�
5. Represent on the number line the intervals 𝑨𝑨 = (−𝟏𝟏, 𝟏𝟏𝟏𝟏] and 𝑩𝑩 = [ 𝟓𝟓, 𝟐𝟐𝟐𝟐). Find the
intervals 𝑨𝑨 ∩ 𝑩𝑩 and 𝑨𝑨 ∪ 𝑩𝑩.
6. Write two intervals which intersection is (−𝟐𝟐, 𝟐𝟐).
7. True or False:
𝟏𝟏 ∈ �𝟎𝟎,
𝟏𝟏
𝟑𝟑
�
𝟒𝟒
𝟗𝟗
∈ �−𝟏𝟏,
𝟏𝟏
𝟐𝟐
�
𝟏𝟏 + √𝟖𝟖 ∈ ( 𝟏𝟏, 𝟑𝟑] √𝟐𝟐
𝟐𝟐
∈ (−𝟏𝟏, 𝟏𝟏)
−𝟓𝟓
𝟒𝟒
∈ [−𝟐𝟐, 𝟏𝟏)
−𝟑𝟑
𝟐𝟐
∈ �
−𝟓𝟓
𝟒𝟒
, 𝟎𝟎�
Axel Cotón Gutiérrez Mathematics 4º ESO Exercises 4.1.3.2