Integers, Absolute Value, and the Coordinate Plane

In everyday life we use integers to express opposite situations. Match an integer (positive or negative) to each of the following situations:

1. a) The street thermometer reads $4$ degrees below zero.
2. b) You owe $\$15$ to a friend.
3. c) A diver is at a depth of $20$ feet.
4. d) An airplane is flying at an altitude of $8.500$ feet.
5. e) You have won $50$ points in a level of your favorite video game.
6. f) The shopping center parking lot is on the third basement level.

Look at the following number line and identify which integer corresponds to each of the points marked with a letter ($A$, $B$, $C$ and $D$).

Look at the following number line where some integers have been replaced by letters. Knowing that the distance between each mark is 2 units:

1. a) Identify the integers that the letters $A$, $B$ and $C$ represent.
2. b) Calculate the absolute value of each of those numbers: $|A|$, $|B|$ and $|C|$.
3. c) Which letter represents the number with the greatest absolute value?

Look at the following number line where two points, $A$ and $B$, have been marked. Knowing that the distance between point $A$ and point $B$ is $15$ units and that point $A$ corresponds to the integer $-6$:

1. a) Determine which integer point $B$ represents if it is located to the right of $A$.
2. b) Calculate the value of $|A| + |B|$.
3. c) If we place a new point $C$ exactly in the middle of the line segment joining $A$ and $B$, what integer would $C$ represent?

The concept of absolute value represents the distance of a number from zero. Work out the following absolute values:

1. a) $|-7| =$
2. b) $|+12| =$
3. c) $|-25| =$
4. d) $|0| =$
5. e) $|-1| =$
6. f) $|+100| =$

The opposite of an integer is one that has the same absolute value but a different sign. Write down the opposite of the following numbers:

1. a) $op(+6) =$
2. b) $op(-9) =$
3. c) $op(-15) =$
4. d) $op(+21) =$
5. e) $op(0) =$
6. f) $op(-1) =$

Order the following mathematical elements from least to greatest, previously calculating their numerical value:

$$-| -12 |, \quad op(+5), \quad | -8 |, \quad op(op(-3)), \quad -(-10), \quad 0, \quad -| -1 |$$

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

1. a) $+5$ ____ $-8$
2. b) $-3$ ____ $-1$
3. c) $0$ ____ $-5$
4. d) $-12$ ____ $-20$
5. e) $+10$ ____ $+2$
6. f) $-1$ ____ $0$

Order the following integers from least to greatest, using the $<$ symbol between them:

$$-5, \quad +3, \quad -1, \quad 0, \quad +7, \quad -8, \quad -2$$

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

1. a) $-15 \quad \square \quad -12$
2. b) $+3 \quad \square \quad -20$
3. c) $|-8| \quad \square \quad |-10|$
4. d) $op(+5) \quad \square \quad op(-2)$

State whether the following statements are true or false and briefly justify your answer:

1. a) Any negative integer is less than zero.
2. b) The absolute value of a number is always positive or zero.
3. c) The opposite of a negative number is another negative number.
4. d) Between two negative numbers, the one with the greater absolute value is larger.
5. e) The number $-5$ is greater than $-2$ because it is further from zero.

Explain whether the following statements are true or false, giving an example or a counterexample in each case:

1. a) If $a$ and $b$ are two negative integers and $|a| < |b|$, then $a > b$.
2. b) The absolute value of an integer is always greater than the number itself.
3. c) The distance between any integer and its opposite is always an even number.
4. d) If we add the absolute value of a number to its opposite, the result is always zero.

Write the integers that simultaneously meet the following three conditions:

1. Its absolute value is greater than $2$ and less than $6$.

2. It is a negative number.

3. When adding its opposite, the result is $0$.

Represent the numbers found on a number line.

I am thinking of an integer $x$ that meets these clues:

• Its absolute value is equal to that of its opposite.

• The number $x$ is greater than $-4$.

• The number $x$ is less than the opposite of $-1$.

• It is not the number zero.

What possible values can $x$ have? Write down all the possible solutions.

Find the secret integer $x$ that simultaneously meets the following four conditions:

• Its absolute value is greater than $5$ and less than $11$: $5 < |x| < 11$.

• The opposite of $x$ is a positive number.

• $x$ is a multiple of $3$.

• When represented on the number line, it is located to the left of $-7$.

Imagine you are in a very large hotel that has floors above and below ground. If you are on floor $-2$ (second basement) and the elevator makes the following movements, which floor will you end up on?

1. Goes up $5$ floors.

2. Goes down $2$ floors.

3. Goes up $1$ floor.

4. Goes down $4$ floors.

Draw a vertical diagram or a number line to help you.

A temperature sensor in a cryogenics laboratory initially reads $12^{\circ}C$. Due to an experiment, the temperature drops by $18^{\circ}C$, then rises by $5^{\circ}C$ and finally drops again by $3^{\circ}C$.

1. a) Represent each of the temperature changes using integers.
2. b) What is the final temperature recorded by the sensor?
3. c) Is the final temperature above or below $0^{\circ}C$?

A delivery driver enters the elevator of a shopping mall on the ground floor ($0$). First he goes up to the warehouse on floor $+5$, then he goes down $8$ floors to collect goods in the garage and, finally, he goes up $2$ floors to make a delivery.

1. a) On which floor is the garage located? Express the result with an integer.
2. b) On which floor does he make the final delivery?
3. c) How many floors has the delivery driver traveled in total during his entire journey? (Hint: use the concept of absolute value).

A volcanic inspection drone takes off from a platform located at $120$ feet above sea level. During its mission, it makes the following consecutive movements:

1. It descends $150$ feet to enter a crevice.

2. It ascends $85$ feet to avoid a gust of wind.

3. It descends again $40$ feet to take thermal samples.

Express each movement as an integer and determine, using a mathematical expression, at what height or depth the drone is relative to sea level at the end of its mission.

The balance of a bank account at the beginning of the month was $\$-245$. During the month the following transactions occur:

• Three equal deposits from a second-hand sales platform worth $\$40$ each.

• A charge for the electricity bill that is double the absolute value of the initial balance.

• A refund for a purchase of $\$120$.

Write a single expression with integers that represents all these transactions and calculate the final balance of the account. Is the account still in the red?

A historian is studying the lives of two figures from Ancient Greece. Figure $A$ was born in the year $342$ BC and figure $B$ was born in the year $125$ BC.

1. a) Represent both years of birth using integers on the number line in your notebook.
2. b) Which of the two figures was born first?
3. c) If figure $B$ lived for $60$ years, in what year did they die? Express it with an integer and indicate whether it was BC or AD.

An archaeologist is studying an ancient dynasty. He knows that the first king of the dynasty was born in the year $42$ BC and died at $65$ years of age. His son, the heir, was born when the father was $28$ years old and reigned for $35$ years after his father's death.

1. a) Use integers to represent the years of birth and death of both figures.
2. b) In what year did the son's reign end? Express the result in chronological format (BC or AD).

In a strategy game, a pirate ship moves across a coordinate map. Its initial position is point $P(-3, 2)$.

1. a) If the ship moves $5$ units to the right, what is its new x-coordinate ($x$)?
2. b) If from that new position it moves down $4$ units, what is its y-coordinate ($y$)?
3. c) Represent the initial point and the final point on a coordinate plane in your notebook.

On the following coordinate plane, a robot starts from point $P(-2, 3)$. It makes three consecutive movements:

1. It moves $5$ units to the right.

2. It moves $7$ units down.

3. It moves $4$ units to the left.

In which quadrant does the robot end up? Do it in your notebook. Indicate the final coordinates of the point and calculate the absolute value of the sum of its new x-coordinate ($x$) and its new y-coordinate ($y$).