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) $(-6) + (+10) =$

Evaluate the following operations by removing the parentheses first:

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

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

Use the sign rules to evaluate these multiplications and divisions:

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

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

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

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

Find the integer that simultaneously satisfies these three conditions:

1. Its absolute value 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, the result is $0$.

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

Evaluate the result of these simple combined operations, respecting the order of operations (remember: multiplications and divisions first, then additions and subtractions):

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

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

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

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

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

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

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) $(-4, -2, +4)$

Evaluate the following combined operations with square brackets. Remember to work out what is inside the brackets first:

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

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

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

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

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

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

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

Place the necessary parentheses in the following expression so that the final result is exactly equal to $0$:

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

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

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

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

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

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

1. a) $x \cdot (-4 + 9) = -25$
2. b) $(-30) : x + (-12) = -18$
3. c) $2 \cdot |x| = 14$ (Find all possible solutions for $x$).

Complete the following "Integer Wall". The number in each upper box must be the result of multiplying the two numbers in the boxes it rests on.

You are on floor $2$ of a shopping mall. You go up $3$ floors to go to the food court and then go down $6$ floors to reach the parking garage. On which floor is the parking garage? Indicate each movement with an integer.

The following movements appear on the bank statement of a comic book store:

• Initial balance: $+450\ \$$

• Payment to supplier: $-620\ \$$

• Sale of collectible figures: $+185\ \$$

• Return of a defective item: $-45\ \$$

• Income from subscriptions: $+110\ \$$

Calculate the final balance of the account. If the bank charges a fee of $15\ \$$ if the account goes into negative numbers (balance less than 0), will the store owner have to pay it?

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 penalized by doubling their negative score accumulated in 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.