"Integers: Opposites, Absolute Value, and Order"

In everyday life, we use integers to express opposite situations with respect to a reference point or "zero point". Match each of the following statements with the integer (positive or negative) that best represents it:

• a) A diver is at a depth of $15$ feet below sea level.

• b) A hiker has climbed to the top of a mountain $2300$ feet high.

• c) The balance on your public transit card is $4$ dollars in debt.

• d) An airplane flies at an altitude of $9500$ feet.

• e) The temperature at a ski resort is $7$ degrees below zero.

Look at the following number line where several points have been marked with letters. Indicate which integer corresponds to each of the letters represented:

Considering that the distance between two consecutive small marks is $1$ unit, determine the values of $A$, $B$, $C$ and $D$.

An underwater archaeology team is exploring the remains of an ancient galleon. They use a submersible drone to document the discovery. Express the following situations with integers:

1. a) The mother ship is on the surface of the sea.
2. b) The galleon is resting on the seabed, at a depth of $45$ feet.
3. c) The drone descends from the surface to a crevice located $12$ feet below the galleon.
4. d) After exploring the crevice, the drone goes up $25$ feet to photograph the ship's mast.

Represent these four positions on a graduated vertical line where the origin is sea level.

The modulus of an integer is the distance that separates it from zero on the number line, without taking its direction into account. Calculate the following moduli:

• a) $|-8| =$

• b) $|+12| =$

• c) $|-35| =$

• d) $|0| =$

• e) $|+7| =$

Two integers are opposites if they are at the same distance from zero but on different sides of the number line. Write the opposite of each of the following numbers and represent them in pairs on a small number line:

• a) $+5$

• b) $-3$

• c) $-1$

• d) $+4$

In a city in the northern United States, the following minimum temperatures were recorded during the first week of January:

• Monday: $-4$ ºF

• Tuesday: $-7$ ºF

• Wednesday: $0$ ºF

• Thursday: $+2$ ºF

• Friday: $-1$ ºF

• Saturday: $+3$ ºF

• Sunday: $-5$ ºF

Order these days from coldest to warmest (that is, from lowest to highest temperature) using the symbol $<$ between the values.

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

• a) $-10 \quad \square \quad -2$

• b) $+5 \quad \square \quad -8$

• c) $-1 \quad \square \quad 0$

• d) $-15 \quad \square \quad -20$

• e) $|-12| \quad \square \quad +10$

Hint: Remember that for negative numbers, the one closer to zero is greater.

Order the following values from smallest to largest using the symbols $<$ or $>$. We recommend calculating the numerical value of each expression first:

$A = |−12|$

$B = op(+7)$

$C = −|−5|$

$D = op(−2)$

$E = −10$

$F = |op(−4)|$

On a set of Cartesian coordinate axes, we have drawn three vertices of a rectangle: $A(−3, 2)$, $B(4, 2)$ and $C(4, −1)$.

1. a) Represent the points on the plane.
2. b) Determine the coordinates of the fourth vertex $D$ to complete the rectangle.
3. c) Which quadrant is each of the vertices in?
4. d) Calculate the perimeter of the rectangle by counting the grid units.

In a strategy video game, the coordinates of your base are $A(-3, 4)$. You must place an outpost at a point $B$ that satisfies the following conditions:

1. Its x-coordinate ($x$-coordinate) is the opposite of the x-coordinate of $A$.

2. Its y-coordinate ($y$-coordinate) is an integer such that its modulus is $2$, but point $B$ must be in a different quadrant from $A$ and cannot be on the x-axis.

Represent both points on the Cartesian plane and indicate which quadrant each one is in. What is the horizontal and vertical displacement needed to go from $A$ to $B$?

An elevator in a very large shopping mall has floors above and below ground. If we consider the ground floor (street level) as the number $0$:

• a) Which floor do we reach if we are in basement $-3$ and we go up $5$ floors?

• b) Which floor do we reach if we are on floor $+2$ and we go down $4$ floors?

• c) If we are in basement $-2$ and we want to go up to floor $+4$, how many floors must we go up?

On a scientific expedition, a drone flies over a volcano at an altitude of $+125$ feet above sea level, while a robotic probe is exploring the bottom of a nearby lake at a depth of $-84$ feet.

1. a) Represent both positions on a vertical number line, placing sea level at the origin.
2. b) Calculate the total distance, in modulus, that separates the drone from the probe.
3. c) If the probe ascends $30$ feet, what will its new position be? Express it as an integer.

Look at the following number line where three points, $X$, $Y$ and $Z$, have been marked.

If each division on the line represents $2$ units:

1. a) Write the coordinates of points $X$, $Y$ and $Z$.
2. b) Calculate the distance between point $X$ and point $Z$.
3. c) Write the opposite of $X$ and represent it on the line.

Complete the following magic pyramid of integers. The rule is very simple: the value of each box is the sum of the two numbers directly below it.

A platform video game scores a character's movements on a horizontal line.

• For each gold coin collected, the character advances $+5$ positions.

• For each obstacle hit, the character moves back $op(+3)$ positions.

• If they fall into a trap, they are placed at position $-|−10|$.

Calculate the character's final position if:

1. They start at the origin ($0$).

2. They collect $4$ gold coins.

3. They hit $2$ obstacles.

4. They fall into a trap.

5. Finally, they advance $8$ more positions.

A content creator analyzes the evolution of their subscribers over a week. On Monday, they started with a base figure. The daily changes were recorded as follows: Tuesday ($+150$), Wednesday (double the loss of Thursday), Thursday ($-200$), Friday (an increase equal to the modulus of Thursday's loss) and Saturday (a drop that left the total weekly balance at exactly $0$ compared to Monday).

Calculate the integer that represents Wednesday's variation and the integer that represents Saturday's variation. Represent the variations for each day on a number line, placing $0$ as the initial state on Monday.

An underwater reconnaissance drone sets off from a platform located at $15$ feet above sea level ($+15$ ft). It makes the following consecutive movements:

• It descends $28$ feet to take samples.

• It ascends a distance equal to half the modulus of its current position.

• Finally, it goes down to an abyssal zone located at $-120$ feet.

Determine the integer that represents the drone's position after the second movement and calculate the total distance traveled by the drone during the three movements.

Consider two integers $a$ and $b$ about which we know the following:

• The modulus of $a$ is $12$.

• The opposite of $b$ is $-15$.

• The distance between $a$ and $b$ on the number line is the smallest possible.

From this data:

1. a) Identify the values of $a$ and $b$.
2. b) What is the largest integer $k$ such that $a < k < b$?
3. c) If we multiply $a$ by a negative number, will the result be greater or less than $b$? Justify your answer.

In a cryogenic laboratory, a sample is at $-18$ °F. The control system allows two types of adjustments to be made:

• Adjustment A: Add $-5$ °F to the current temperature.

• Adjustment B: Change the temperature to its opposite and then subtract $10$ °F.

If we apply Adjustment B first and then Adjustment A, what will the final temperature be? Would the result be the same if we applied Adjustment A first and then Adjustment B? Justify your answer by carrying out both calculations step by step.

Write all the integers $x$ that satisfy the following conditions at the same time:

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

2. They are numbers less than $2$ ($x < 2$).

3. They are numbers greater than or equal to $-3$ ($x \geq -3$).

Represent them on a number line.

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

1. Their modulus is less than $10$, that is: $|x| < 10$.

2. It is a number strictly less than its opposite: $x < Op(x)$.

3. When adding $-3$ to it, the result is a number whose modulus is greater than $5$.

List the numbers obtained and represent them on the number line.

You have three consecutive integers whose sum is a negative number. If the modulus of the middle number is $0$:

1. a) What are those three numbers?
2. b) Order the moduli of these numbers from smallest to largest.
3. c) What happens to the sign of the sum if we choose the three "largest" consecutive integers (furthest to the right on the number line) that are still all negative?

Indicate whether the following statements are true or false. If false, give an example (counterexample) to prove it:

1. a) The modulus of a number is always greater than the number itself.
2. b) If two numbers are negative, the one with the smaller modulus is greater.
3. c) The opposite of an integer is always a negative number.
4. d) The distance between a number and its opposite is always double its modulus.

Analyze the following statements and determine whether they are true or false, providing a counterexample for those you consider false:

1. If $|a| > |b|$, then it is always true that $a > b$.

2. The modulus of the sum of two integers is always equal to the sum of their moduli: $|a + b| = |a| + |b|$.

3. If $n$ is a negative integer, then its opposite $Op(n)$ is always greater than $n$.

4. The distance between two opposite numbers is always double the modulus of one of them.