## Math 6 Chapter 2 Lesson 2: Set of Integers

## 1. Summary of theory Tóm

### 1.1. The set of integers

\(\mathbb{Z} = {\rm{\{ }}\underbrace {… {\rm{; – 3; – 2; – 1;}}}_{nguyen\,\,am}\underbrace {{\rm{0;}}}_{so\,\,0}\underbrace {{\rm{1;2;3;}}…}_{nguyen\,\,duong}{\rm {\} }}\)

-3; -2; -1 are negative integers

1, 2, 3 are positive integers (non-zero natural numbers)

The number 0 is an integer that is neither positive nor negative.

Horizontal axis represents integers

-1 and 1 are opposite numbers

General: a and –a are opposite numbers. Two points representing two opposite numbers are symmetrical about zero.

**Attention:**

+ \(N \subset \mathbb{Z}\). Especially \(N = {\mathbb{Z}_ + }\) (positive integers).

+ The numbers \(a \ge 0\) are called non-negative numbers. a > 0 is a positive number.

+ The numbers \(a \le 0\) are called non-negative numbers. a < 0 is a negative number.

Order in \(\mathbb{Z}\)

– Every non-negative number is greater than every negative number.

1 > – 1000; 0 > – 2012

– Integer a is smaller than integer b (a < b), then the point representing the number a is to the left of the point representing the number b on the number line.

### 1.2. Absolute value of integer

\(\left| A \right| = \left\{ \begin{array}{l}A\,\,neu\,\,A\, \ge 0\\ – A\,neu\,\,A \, < 0\end{array} \right.\)

\(A\,\,neu\,\,A\, \ge 0\) (ie the absolute value of a positive number is itself)

– A if A < 0 (the absolute value of a negative number is its reciprocal)

**Attention: ** The absolute value of a number a is always non-negative.

Write:

|+3| = -|3| = 3: Two opposite numbers have the same absolute value.

|x| = -1 makes no sense.

\(\left| a \right|{\rm{ }} = {\rm{ }}4 \Rightarrow a = \pm 4\) Special |0| = 0

## 2. Illustrated exercise

**Question 1:** Find \(x \in \mathbb{Z}\) such that:

a) -4 < x < 2 b) -2 < x < 2 c) |x| < 3

d) -3 < |x| \( \le 4\) e) |x| > 5.

__Solution guide__

a) \(x \in {\rm{\{ – 3; – 2; – 1;0;1\} }}\)

b) \(x \in {\rm{\{ }} – 1;0;1\} \)

c) \(|x|\,\, < \,\,3 \Rightarrow - 3 < x < 3 \Rightarrow x \in {\rm{\{ }} - 2; - 1;0;1;2\ } .\)

d) \( – 3 < \,\,|x|\,\, \le 4\,\, \Rightarrow \,x \in {\rm{\{ }} - 4; - 3; - 2; - 1;0;1;2;3;4\} \)

e) \(|x|\,\,\, > 5 \Rightarrow x \in {\rm{\{ }}… {\rm{; – 8; – 7; – 6;6;7;8 ;}}… {\rm{\} }}\)

**Verse 2: **Find \(x \in \mathbb{Z}\) such that:

a) |x| = 9 and x < 0 b) |x| = 5

c) |x| = -12 d) |x| = |-2012|

__Solution guide__

a) \(\left| x \right|{\rm{ }} = {\rm{ }}9 \Rightarrow x = \pm 9\) combined with x < 9 , we get x = - 9.

b) \(\left| x \right|{\rm{ }} = {\rm{ }}5 \Rightarrow x = \pm 5\)

c) \(\left| x \right|{\rm{ }} = {\rm{ }} – 12 \Rightarrow x = \emptyset \,\,\)because \(|x|\,\, \ge \,\,0\) for all \(x \in \mathbb{Z}\)

d) \(\left| x \right|{\rm{ }} = {\rm{ }}\left| { – 2012} \right| = |2012|\, \Rightarrow x = \pm 2012.\)

**Question 3: **Calculate

a) (|-24| : |-8|) – 1

b) (|1440| : |-32|) : |-5|.

__Solution guide__

a) |-24| = 24, |-8| = 8

so (|-24| : |-8|) – 1 = (24 : 8) – 1 = 3 – 1 = 2.

b) (|1440| : |-32|) : |-5| = (1440 : 32) : 5 = 45 : 5 = 9.

## 3. Practice

### 3.1. Essay exercises

**Question 1: ** Find \(x,{\rm{ }}y \in \mathbb{Z}\) such that

a) |x| + |y| = 4.

b) \(\left| x \right|{\rm{ }} + {\rm{ }}\left| y \right|\,\,\, \le \,\,2\)

**Verse 2: **Prove that for any \(a \in \mathbb{Z}\), we always have:

a) \(|a| + a \ge 0\) b) \(|a| – a \ge 0.\)

**Question 3: **a) Find x so that |x – 1| + 2012 reached the minimum value.

b) Find x, y \( \in \mathbb{Z}\) knowing that \(|x| + |y|\,\, \le 0\)

### 3.2. Multiple choice exercises Bài

**Question 1: **The set of integers is denoted by:

A. WOMEN

B. N*

C. Z

D.Z*

**Verse 2:** The reciprocal of -3 is:

A. 3

B. -3

C. 2

D. 4

**Question 3: **Choose the wrong sentence?

A. N Z

B. N* Z

C. Z = {…; -2; -first; 0; first; 2; …}

D. Z = {…; -2; -first; first; 2; …}

**Question 4: **Select the correct answers:

A. -6 FEMALE

B. 9 Z

C. -9 FEMALE

D. -10 Z

**Question 5:** The point that is three units away from -1 in the negative direction is:

A. 3

B. -3

C. -4

D. 4

## 4. Conclusion

Through this lesson The collection of integers, you need to complete some of the objectives given by the lesson, such as:

- Integer concept
- Absolute value of integer
- Apply theory to do some exercises related to the set of integers

.

=============

### Related posts:

- Math 6 Chapter 2 Lesson 9: Triangles
- Math 6 Chapter 2 Lesson 8: Circles
- Math 6 Chapter 2 Lesson 7: Practice measuring angles on the ground
- Math 6 Chapter 2 Lesson 6: Bisector of angle
- Math 6 Chapter 2 Lesson 5: Draw an angle that gives the measure
- Math 6 Chapter 2 Lesson 4: When is angle xOy + angle yOz= angle xOz?
- Math 6 Chapter 2 Lesson 3: Angle measure
- Math 6 Chapter 2 Lesson 2: Angle
- Math 6 Chapter 2 Lesson 1: Half plane

## Leave a Reply