The Set of Integers

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

1. a) You find a chest and gain $450$ points.
2. b) An enemy defeats you and you 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.

Match an integer to each of the following situations taken from the everyday life of a teenager:

1. a) The balance on your transit card after having spent $\$5$ on trips and having a previous top-up of $\$2$.
2. b) The position of an exploration submarine in a video game that is $150$ feet below sea level.
3. c) The temperature at a ski resort that shows $4$ degrees below zero.
4. d) The year in which the abacus is estimated to have been invented, approximately $500$ years BC.
5. e) Taking the elevator in a shopping mall from the movie theater floor (floor $3$) to the parking lot located two floors below.

Observe 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 notebook and represent the following integers on it:

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

You are in a very large shopping mall that has floors above and below street level (floor $0$).

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

Compare the following pairs of numbers using the signs $<$ (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) $-1 \quad \dots \quad +1$

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 notebook, filling in the empty gaps 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 where several points have been marked with letters. Identify the integer that corresponds to each letter and, subsequently, calculate the modulus of each of them.

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 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?

A research submarine is at a depth of $-450$ feet with respect to sea level. It launches a vertical probe that ascends $125$ feet to take a sample, and then descends triple what it had ascended.

1. a) Express each of the movements 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).

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

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

In a high mountain city, the meteorological service records the following temperatures throughout a winter day:

• At 6:00 a.m.: $-8$ ºC.

• At 12:00 p.m.: The temperature has risen by $10$ ºC compared to 6:00 a.m.

• At 8:00 p.m.: The temperature has dropped by $5$ ºC compared to 12:00 p.m.

• At 12:00 a.m.: The temperature is the opposite of that recorded at 12:00 p.m.

Represent these four temperatures on a number line and order them from smallest to largest using the $<$ sign.

A toy drone moves along a straight corridor that we have marked as a number line. The take-off point is $0$. The drone makes the following consecutive movements:

1. It moves forward $+6$ feet.

2. It moves backward $10$ feet.

3. It moves to the opposite point of its current position.

4. Finally, it moves forward 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.

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 this is the opposite of the sum of the three previous terms.

Write all the integers $x$ that simultaneously satisfy the two conditions indicated below:

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

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

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

• Their modulus 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 the set of solutions obtained on the number line.

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 $T$ that satisfies:

• $|T| < 15$

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

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

What was the maximum temperature recorded? Indicate all possibilities if there are any.

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 $1$ foot.
3. c) What is the symmetrical point of vertex $A$ with respect to the x-axis (horizontal axis)? Indicate its coordinates as an integer.