Exercises: The set of integers

Express the following real-life situations related to social media and the daily life of a teenager with an integer ($+$ or $-$):

1. a) You have gained $150$ new followers on your TikTok account this week.
2. b) A friend owes you $\$12$ for the birthday present you bought together.
3. c) The street thermometer reads $4$ degrees below zero.
4. d) You have gone down to the second basement of the shopping center to leave your bicycle.

Given the following list of numbers, classify them by indicating which belong to the set of natural numbers ($\mathbb{N}$) and which are integers ($\mathbb{Z}$) but not natural numbers:

Look at the following number line and identify which integer corresponds to each of the points marked with the letters $A$, $B$ and $C$.

A drone is flying at an altitude of $+35$ feet above sea level, while a diver is exploring a coral reef at a depth of $-15$ feet.

1. a) Write down the absolute value of both numbers: $|+35|$ and $|-15|$.
2. b) Which of the two is at a greater distance from sea level ($0$ feet)?
3. c) Write down the opposite of the diver's position.

At a ski resort, the following minimum temperatures have been recorded over four days. Order them from lowest to highest using the $<$ symbol:

• Monday: $-6$ $^\circ C$

• Tuesday: $+2$ $^\circ C$

• Wednesday: $-9$ $^\circ C$

• Thursday: $0$ $^\circ C$

In a large shopping mall, the elevator buttons represent the floors. The ground floor is $0$, the upper floors are positive ($+1, +2...$) and the parking garage basements are negative ($-1, -2...$).

1. a) If you are on floor $-3$ and go up to the opposite floor, which floor do you arrive at?
2. b) What is the absolute value of the number of floor $-2$?
3. c) Which floor is further from the street (floor $0$): $+4$ or $-5$?

Identify the Cartesian coordinates $(x, y)$ of the points $P$ and $Q$ represented on the following grid. Remember that the first number is the horizontal axis (x-axis) and the second is the vertical axis (y-axis).

Write down the integer that represents the final state of a young person's bank account in these two independent situations:

1. a) They had $\$25$ saved and bought a video game that cost $\$30$ (the bank allows them to go into negative balance).
2. b) They owed their sister $\$10$ and their grandfather gave them a $\$10$ bill to settle the debt.

An underwater exploration drone performs the following maneuvers starting from sea level (which we represent as the value $0$):

• It ascends $12$ feet to calibrate its signal.

• It descends $25$ feet to take coral samples.

• It goes up $8$ feet to avoid a strong current.

• It goes down another $10$ feet to enter an underwater cave.

Indicate the final position of the drone with respect to sea level using integers and draw a vertical diagram in your notebook to represent these positions.

Look at the following number line where four points have been marked: $A$, $B$, $C$ and $D$. Identify the integer that corresponds to each letter knowing that point $C$ represents the number $-2$ and that the distance between each small mark is one unit.

In the physics laboratory, the freezing temperatures of various liquids are being analyzed. The results are:

• Liquid P: $-8$ ºF

• Liquid Q: $-15$ ºF

• Liquid R: $-3$ ºF

• Liquid S: $-12$ ºF

Order the liquids from lowest to highest temperature using the $<$ symbol and explain which of them has the greatest absolute value.

Write down all the integers $x$ that simultaneously satisfy these two conditions:

1. Their absolute value is less than $6$ ($|x| < 6$).

2. It is a number strictly less than $-2$ ($x < -2$).

Represent the solution on a number line.

An archaeologist finds a coin at an excavation site. Upon investigation, they discover that it was minted in the year $45$ BC. If the archaeologist lives in the year $2024$ AD, answer:

1. a) What integer would you assign to each date?
2. b) What is the opposite of the year the coin was minted?
3. c) Represent both years approximately on a timeline.

Compare the following pairs of integers using the $>$ or $<$ signs and justify your answer based on their position on the number line:

1. a) $-15$ _____ $-20$
2. b) $+3$ _____ $-100$
3. c) $0$ _____ $-5$
4. d) $op(+7)$ _____ $-6$

Two divers, Emma and Daniel, are in the ocean. Emma is at a depth of $18$ feet below sea level. Daniel is $12$ feet from Emma, but he is in a deeper position.

1. a) At what depth is Daniel? Express this as an integer.
2. b) If Emma goes up $5$ feet, what will her new position be?
3. c) What is the absolute distance between Emma's new position and Daniel's position?

In a board game, your counter is on the square that represents the integer $-4$. You roll a special die and get the following consecutive moves:

1. You move forward $6$ spaces.

2. You move backward double what the absolute value of $-3$ indicates.

3. You move to the opposite of the square you are on at that moment.

Which square do you finish the game on? Indicate all the intermediate steps.

On a scientific expedition, an exploration drone and a mini-submarine set off from the deck of a ship that is exactly at sea level (altitude $0$ feet).

1. The drone ascends $145$ feet, then descends $60$ feet and finally goes back up $25$ feet.

2. The submarine descends $85$ feet, goes up $30$ feet and goes back down until it reaches a depth that doubles its previous distance from sea level.

If at that moment the drone drops a probe that travels through the air and water until it reaches the submarine, what total distance does the probe travel? Represent the situation on a vertical axis.

<line x1="0" y1="310" x2="400" y2="310"/>

<line x1="0" y1="370" x2="400" y2="370"/>

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

1. Their absolute value is greater than $4$ but less than $10$.

2. The opposite of $x$ is a negative number.

3. If you add its absolute value to $x$, the result is less than $16$.

We are designing a video game where a character must jump on platforms numbered with integers. The character starts on platform $0$.

• The first jump is $+5$ units.

• The second jump is twice as long as the first, but in the opposite direction.

• The third jump takes them to a platform whose number is the opposite of the number of the platform they were on after the second jump.

• The fourth jump is $-12$ units.

Which platform are they on at the end? Draw the path on the number line indicating each step.

Three cities have very curious extreme temperatures on a winter's day:

• City A has a temperature that is the opposite of $6$.

• City B has a temperature such that its absolute value is $4$, and it is warmer than city A.

• City C has a temperature that is exactly the same distance from the temperature of A as it is from the temperature of B.

Calculate the temperature of the three cities and order them from lowest to highest using the $<$ symbol.

On a Cartesian plane, a square has two of its opposite vertices at the points $A(-2, 3)$ and $C(4, -3)$.

1. Draw the axes and plot the points $A$ and $C$.

2. Determine the coordinates of the other two vertices, $B$ and $D$, knowing that the square is not rotated (its sides are parallel to the axes).

3. What are the coordinates of the central point where the diagonals of the square intersect?

A historian is analyzing the life of an ancient philosopher. The data they have is as follows:

• The philosopher was born in the year $42$ BC (which we represent as $-42$).

• His son was born when the philosopher was $31$ years old.

• The philosopher lived $12$ years longer than his son.

• The son died in the year $28$ AD (which we represent as $+28$).

How many years did the philosopher live? In what year did he die? Is it possible that the philosopher knew his grandson if he was born the year the son died? Justify your answer using integers.

Imagine two integers, $a$ and $b$. We know that:

1. The distance between $a$ and $0$ is $7$ units.

2. The distance between $a$ and $b$ is $3$ units.

3. $b$ is less than $a$.

Determine all the possible values for $a$ and for $b$. For each case, write down the absolute value of the sum $|a + b|$.

A bank account has had the following movements over the last five days:

• Monday: Deposit of an unknown amount of money $X$.

• Tuesday: Payment of a bill of $\$120$.

• Wednesday: Collection of an invoice of $\$200$.

• Thursday: The balance tripled compared to the previous day (Wednesday).

• Friday: $\$150$ was withdrawn and the final balance was exactly $\$240$.

What was the account balance on Monday before the movements started? Set up the situation using a chain of operations with integers.