The set of integers

In a survival video game session, your character's actions affect their experience points. Express the following situations using an integer:

1. a) You find a chest and gain $450$ points.
2. b) You are defeated by an enemy and lose $125$ points.
3. c) You fall into a trap and your score drops by $50$ points.
4. d) You complete a side quest and receive $80$ points.

State whether the following statements are true or false:

1. a) The number $-7$ is a natural number ($\mathbb{N}$).
2. b) All natural numbers are also integers ($\mathbb{Z}$).
3. c) The absolute value of a number is always positive or zero.
4. d) The opposite of a negative number is always negative.

Compare the following pairs of numbers using the symbols $<$ (less than) or $>$ (greater than):

1. a) $-8 \quad \dots \quad -2$
2. b) $+5 \quad \dots \quad -10$
3. c) $0 \quad \dots \quad -3$
4. d) $-1 \quad \dots \quad +1$

Consider two integers $a$ and $b$. It is known that $a$ is a negative number and $b$ is a positive number. State whether the following statements are always true, sometimes true, or always false, justifying your answer with an example:

1. a) $|a|$ is always greater than $b$.
2. b) The opposite of $b$ is always less than $a$.
3. c) The absolute value of $a$ is always a positive number.
4. d) $b$ is always greater than $a$.

Write down all the integers $x$ that satisfy both of the following conditions simultaneously:

1. Their absolute value is less than or equal to $4$: $|x| \le 4$.

2. They are greater than the opposite of $2$: $x > Op(+2)$.

Order the following numbers from smallest to largest, placing the negative values first and using the corresponding inequality signs:

$$|-7|, \quad +3, \quad Op(-5), \quad -4, \quad 0, \quad Op(+8), \quad -1, \quad |+2|$$

Determine all the integers $x$ that simultaneously satisfy the following three conditions:

• Its absolute value is greater than $5$ and less than or equal to $9$.

• The number $x$ does not belong to the set of natural numbers $\mathbb{N}$.

• $x$ is an even number.

Graphically represent on the number line the set of solutions you obtained. You can use this unmarked number line as a guide:

You can do it on paper and send a photo with your answer.

On a coordinate grid, three of the vertices of a rectangle are the points $A(-4, 3)$, $B(2, 3)$ and $C(2, -1)$.

1. a) Determine the coordinates of the fourth vertex $D$ to complete the rectangle.
2. b) Calculate the perimetre of the rectangle knowing that each grid unit is equivalent to $1$ metre.
3. c) What is the symmetric point of vertex $A$ with respect to the x-axis (horizontal axis)? State its coordinates using integers.