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.

Calculate the absolute value and the opposite of the following integers:

1. a) $-14$
2. b) $+21$
3. c) $-5$
4. d) $+9$

Complete the following table in your exercise book, filling in the blanks so that each row is consistent with the definition of an integer, its opposite, and its absolute value:

Number ($z$)	Opposite ($Op(z)$)	Absolute Value ($|z|$)
$-12$
	$+7$
		$5$ (If $z$ is positive)
		$9$ (If $z$ is negative)
$Op(-3)$

Look at the following number line and state which integers correspond to the points marked with the letters $A$, $B$, $C$, and $D$.

Draw a number line in your exercise book and represent the following integers on it:

$$-5, \quad +3, \quad -2, \quad 0, \quad +1, \quad -4$$

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

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) $-1 \quad \dots \quad +1$

A group of friends is measuring the temperature in different European cities on a winter day:

• Berlin: $-4 \text{ °C}$

• Madrid: $+9 \text{ °C}$

• Oslo: $-12 \text{ °C}$

• Rome: $+15 \text{ °C}$

• Zurich: $-1 \text{ °C}$

Order the cities from coldest to warmest.

Look at the following number line where several points have been marked with letters. Identify the integer corresponding to each letter and then calculate the absolute value of each of them.

In a high-mountain town, the weather service records the following temperatures throughout a winter's day:

• At 06:00: $-8$ ºC.

• At 12:00: The temperature has risen by $10$ ºC compared to 06:00.

• At 20:00: The temperature has fallen by $5$ ºC compared to 12:00.

• At 00:00: The temperature is the opposite of the one recorded at 12:00.

Represent these four temperatures on a number line and order them from lowest to highest using the $<$ symbol.

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 $a$.

Reason whether the following statements are true or false, providing a counterexample if they are false:

1. a) If an integer $a$ is less than $b$, then the modulus of $a$ is always less than the modulus of $b$ ($|a| < |b|$).
2. b) The opposite of the modulus of any integer (other than zero) is always a negative number.
3. c) There is no integer whose modulus is equal to its opposite.

Given the integers $a = -12$ and $b = +5$:

1. a) Calculate the distance between them on the number line.
2. b) Find the number $c$ that is at exactly the same distance from $a$ as from $b$. Is $c$ an integer? Justify your answer.
3. c) If $d$ is a number such that $|d| = |a| + |b|$, what values can $d$ take?

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:

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

A toy drone moves along a straight corridor marked as a number line. The takeoff point is $0$. The drone makes the following consecutive moves:

1. Moves forward $+6$ meters.

2. Moves back $10$ meters.

3. Moves to the point opposite its current position.

4. Finally, advances to a point whose absolute value is $2$ and which is to the left of the origin.

At what point on the line is the drone at the end of its journey? Draw it in a diagram.

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.

Imagine that a smart thermometer records the maximum and minimum temperature of a city. One day, the minimum temperature was $-8$ $^\circ C$. The maximum temperature was an integer $T$ satisfying:

• $|T| < 15$

• The difference between the maximum and minimum (temperature range) is $12$ degrees.

• $T$ is a number greater than its opposite.

What was the maximum recorded temperature? State all possibilities, if any.

Associate an integer with each of the following situations taken from a teenager's daily life:

1. a) The balance on your transit card after spending $100$ pesos on trips and having a previous reload of $40$ pesos.
2. b) The position of an exploration submarine in a video game that is $150$ meters below sea level.
3. c) The temperature at a ski resort reading $4$ degrees below zero.
4. d) The year in which the abacus is estimated to have been invented, approximately $500$ years BC.
5. e) Going down in a shopping center elevator from the movie theater floor (floor $3$) to the parking lot located two floors below.

You are in a very large shopping center with floors above and below street level (floor $0$).

1. a) If you are on floor $-3$ and go up $5$ floors, which floor are you on?
2. b) If you are on floor $2$ and go down $3$ floors, which floor do you reach?

A research submarine is at a depth of $-450$ meters relative to sea level. It launches a vertical probe that ascends $125$ meters to take a sample, and subsequently descends three times the distance it had ascended.

1. a) Express each movement and the final position of the probe using integers.
2. b) Calculate the absolute distance between the initial position of the submarine and the final position of the probe.

Write the name of a historical figure who satisfies the following conditions:

• They were born in the year $x$, where $x$ is the opposite of $356$.

• They died in the year $y$, where $y$ is an integer such that $|y| = 323$ and $y < 0$.

• Calculate how many years this figure lived, knowing that there is no year zero in the Gregorian calendar (it goes from 1 BC to 1 AD).

A sequence of integers is defined as follows:

1. The first term is $a_1 = -5$.

2. The second term is the opposite of the modulus of $a_1$, increased by $2$ units.

3. The third term is the modulus of the second term, decreased by $10$ units.

Calculate the fourth term if it is the opposite of the sum of the three previous terms.

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 perimeter of the rectangle knowing that each grid unit is equivalent to $1$ meter.
3. c) What is the symmetric point of vertex $A$ with respect to the x-axis (horizontal axis)? State its coordinates using integers.