Operations with integers

Calculate the following simple additions and subtractions of integers:

1. a) $(+7) + (+5) =$
2. b) $(-8) + (-3) =$
3. c) $(+9) + (-4) =$
4. d) $(-6) + (+10) =$

In Valle Nuevo, the thermometer read $-4$ ºC at seven in the morning. If by midday the temperature has risen by $9$ ºC, what temperature does the thermometer show now? Express the calculation using integers.

Solve the following operations by removing the brackets first:

1. a) $12 + (-5) =$
2. b) $8 - (+10) =$
3. c) $(-3) - (-7) =$
4. d) $(-5) + (+5) =$

A diver is at a depth of $-15$ meters relative to sea level. If he decides to descend another $12$ meters, what depth will he be at now? Draw this on the vertical number line.

You are on floor $2$ of a shopping center. You go up $3$ floors to get to the food court and then go down $6$ floors to reach the parking lot. Which floor is the parking lot on? State each movement using an integer.

Nadia has $25$ pesos in her bank account. If the bank charges her an internet bill of $30$ pesos, what will her new balance be? Express the result as an integer.

A weather station in Valle Nuevo recorded the following temperature changes over the course of a winter's day:

1. At 8:00, the temperature was $-5\ ^\circ C$.

2. At 12:00, it had risen by $8\ ^\circ C$.

3. At 18:00, it dropped by $4\ ^\circ C$ compared to the previous reading.

4. At 22:00, it dropped again by another $6\ ^\circ C$.

Write a single mathematical expression using integers that reflects all these changes, and calculate the final temperature at the end of the day.

Use the rule of signs to solve these multiplications and divisions:

1. a) $(+4) \times (-3) =$
2. b) $(-5) \times (-2) =$
3. c) $(-20) : (+5) =$
4. d) $(-18) : (-3) =$

Find the integer that simultaneously satisfies these three conditions:

1. Its modulus is greater than $5$ and less than $10$.

2. When multiplied by $-3$, the result is a positive number.

3. If you add its opposite to it, the result is $0$.

Is there more than one possible solution? Justify your answer.

Apply the distributive property to calculate these expressions in the simplest way possible (factorise out a common factor when necessary):

1. a) $(-7) \times (+12) + (-7) \times (-2)$

1. b) $15 \times (-4) - 15 \times (+6)$

1. c) $(-10) \times (-3) + (-10) \times (+13)$

A "trio of integers" is said to be balanced if the sum of the first two multiplied by the third results in $-24$.

Determine which of the following trios are balanced by performing the corresponding operations:

1. a) $(+5, -8, +8)$
2. b) $(-10, +2, +3)$
3. c) $(-4, -2, +4)$

Calculate the result of these simple combined operations, respecting the order of operations (remember: multiplication and division first, then addition and subtraction):

1. a) $5 + 2 \times (-3) =$
2. b) $(-10) : 2 + 8 =$
3. c) $4 \times 3 - 15 =$

Calculate the result of the following combined operations, paying special attention to the order of operations and the rule of signs:

1. a) $15 - (8 - 12) + (-5) \times 2$

1. b) $(-3) \times [(-2) + (-4) \times (-3)]$

1. c) $24 : (-4) - (-6) : (+3)$

1. d) $(-10) + (-2) \times (-5) + (-20) : 5$

Solve the following combined operations with square brackets. Remember to perform the operations inside the brackets first:

1. a) $18 - [ 2 + (-5) - (4 - 7) ]$

1. b) $[ (-6) \times 2 - (-10) ] : (-2)$

1. c) $30 : [ (-4) + (-2) \times (-3) ]$

1. d) $(-5) \times [ 8 - (12 : (-3)) ]$

Calculate the value of the following combined expression step by step, strictly respecting the order of operations:

$(-2)^4 + \sqrt{25} \times (-3) - [(-8 + 10)^3 : (-2^2)]$

Note: Pay attention to the difference between $(-2)^2$ and $-2^2$.

An inspection drone starts at a height of $15$ meters above bridge level. It makes the movements shown in the diagram (climbing in orange, descending in gray):

Using an addition of integers, calculate the final position of the drone relative to the bridge. Is it above or below it?

On the bank statement of a comic book shop, the following transactions appear:

• Starting balance: $+\text{RD\$}\ 27\,000$

• Payment to supplier: $-\text{RD\$}\ 37\,200$

• Sale of collectible figures: $+\text{RD\$}\ 11\,100$

• Refund for a defective item: $-\text{RD\$}\ 2\,700$

• Income from subscriptions: $+\text{RD\$}\ 6\,600$

Calculate the final balance of the account. If the bank charges a fee of $\text{RD\$}\ 900$ if the account goes overdrawn (balance less than 0), will the shop owner have to pay it?

A marine research team uses sonar to measure the distance between a rescue helicopter and a submarine conducting depth tests.

If the helicopter is at a height of $+120$ meters and the submarine is at a depth of $-85$ meters:

1. a) What is the total vertical distance separating them?
2. b) If the submarine ascends $25$ meters and the helicopter descends $30$ meters, what will the new distance between them be? Express it using operations with integers.

A caving team is exploring a deep chasm. They start at the entrance (level $0$ meters). They make the following moves consecutively:

1. They descend $45$ meters.

2. They climb up $12$ meters to set up a base camp.

3. They descend three times as far as they have just climbed.

4. Finally, they descend another $18$ meters to the bottom.

Write a single expression using integers reflecting the whole process and calculate the depth of the bottom of the chasm.

In a strategy video game, a player starts with $500$ honor points. The rules are:

• For each battle won: $+120$ points.

• For each battle lost: $-150$ points.

• If they lose three battles in a row, they are penalised by doubling their accumulated negative score from those three battles.

If a player wins $2$ battles and then loses $3$ battles in a row, write down the combined operation to calculate their final score and solve it.

Find the unknown integer $x$ in each of the following equalities:

1. a) $x \times (-4 + 9) = -25$

1. b) $(-30) : x + (-12) = -18$

1. c) $2 \times |x| = 14$ (Find all possible solutions for $x$).

Place the necessary parentheses and brackets in the following expression so that the final result is exactly equal to $-5$:

$-15 + 3 \times -4 - 12 : -6 + 1$

Complete the following "Integer Wall". The number in each upper cell must be the result of multiplying the two numbers in the cells supporting it.

Make up a real-life problem that can be solved using the following mathematical expression and then calculate the final result to answer your problem:

$100 + 4 \times (-15) - 3 \times (10 - 4)$