✓ On a number line, integers to the right are always greater

✓ On a number line, integers to the left are always smaller

✓ All positive integers are greater than zero and all negative integers

✓ All negative integers are less than zero and all positive integers

Ordering: \(-6 < -5 < -4 < -3 < -2 < -1 < 0 < 1 < 2 < 3 < 4 < 5 < 6\)

Example 1: \(-5 < -2\) (because -5 is to the left of -2)

Example 2: \(3 > -8\) (all positive integers are greater than negative)

Example 3: Order from least to greatest: \(\{-3, 5, -7, 0, 2\}\)
Answer: \(-7, -3, 0, 2, 5\)

Rule 1: Positive + Positive = Positive
Formula: \((+a) + (+b) = +(a + b)\)
Example: \((+5) + (+3) = +8\)

Rule 2: Negative + Negative = Negative
Formula: \((-a) + (-b) = -(a + b)\)
Example: \((-5) + (-3) = -8\)

Rule 3: Positive + Negative (or vice versa)
Formula: \(a + (-b) = a - b\) (subtract and keep sign of larger absolute value)
Example: \(7 + (-4) = +3\) (larger absolute value is 7)
Example: \(4 + (-7) = -3\) (larger absolute value is 7)

Master Rule: Change subtraction to addition and flip the sign of the second number
"Keep-Change-Change" or "Same-Change-Change"

Formula: \(a - b = a + (-b)\)

Rule 1: Positive - Positive
Formula: \((+a) - (+b) = a + (-b)\)
Example: \(5 - 8 = 5 + (-8) = -3\)

Rule 2: Positive - Negative
Formula: \(a - (-b) = a + (+b) = a + b\)
Example: \(6 - (-4) = 6 + 4 = 10\)

Rule 3: Negative - Positive
Formula: \(-a - (+b) = -a + (-b) = -(a + b)\)
Example: \(-5 - 7 = -5 + (-7) = -12\)

Rule 4: Negative - Negative
Formula: \(-a - (-b) = -a + (+b)\)
Example: \(-8 - (-3) = -8 + 3 = -5\)

Positive Counters (⊕): Represent positive integers

Negative Counters (⊖): Represent negative integers

Zero Pair: One positive counter + one negative counter = 0

Step 1: Use counters to represent each integer

Step 2: Combine or remove counters based on operation

Step 3: Create zero pairs (cancel out)

Step 4: Count remaining counters for the answer

Example: \(5 + (-3)\) → 5 positive counters + 3 negative counters
Create 3 zero pairs → 2 positive counters remain → Answer: \(+2\)

Step 1: Change all subtractions to addition (flip signs)

Step 2: Group positive integers together

Step 3: Group negative integers together

Step 4: Add within each group

Step 5: Combine the two groups using addition rules

Example 1: \(7 + (-3) + 5 + (-9)\)
Group positives: \(7 + 5 = 12\)
Group negatives: \((-3) + (-9) = -12\)
Combine: \(12 + (-12) = 0\)

Example 2: \(-4 + 8 - 5 + 2\)
Rewrite: \(-4 + 8 + (-5) + 2\)
Group positives: \(8 + 2 = 10\)
Group negatives: \(-4 + (-5) = -9\)
Combine: \(10 + (-9) = 1\)

Step 1: Identify what the question is asking

Step 2: Assign positive or negative values to quantities

Step 3: Determine if you need to add or subtract

Step 4: Write the expression

Step 5: Calculate using integer rules

• Temperature: Above zero = positive, Below zero = negative

• Money: Earning/deposits = positive, Spending/withdrawals = negative

• Elevation: Above sea level = positive, Below sea level = negative

• Football: Gain yards = positive, Loss yards = negative

Problem: The temperature was \(-5°C\) in the morning. It rose \(8°C\) by afternoon. What is the afternoon temperature?
Solution: \(-5 + 8 = +3°C\)

SAME SIGNS → POSITIVE RESULT

DIFFERENT SIGNS → NEGATIVE RESULT

Rule 1: Positive × Positive = Positive
Formula: \((+a) \times (+b) = +(a \times b)\)
Example: \((+6) \times (+4) = +24\)

Rule 2: Negative × Negative = Positive
Formula: \((-a) \times (-b) = +(a \times b)\)
Example: \((-6) \times (-4) = +24\)

Rule 3: Positive × Negative = Negative
Formula: \((+a) \times (-b) = -(a \times b)\)
Example: \((+6) \times (-4) = -24\)

Rule 4: Negative × Positive = Negative
Formula: \((-a) \times (+b) = -(a \times b)\)
Example: \((-6) \times (+4) = -24\)

Step 1: Find the absolute values (ignore signs)

Step 2: Multiply the absolute values

Step 3: Apply the sign rule to determine final sign

SAME SIGNS → POSITIVE RESULT

DIFFERENT SIGNS → NEGATIVE RESULT

Rule 1: Positive ÷ Positive = Positive
Formula: \((+a) \div (+b) = +(a \div b)\)
Example: \((+24) \div (+6) = +4\)

Rule 2: Negative ÷ Negative = Positive
Formula: \((-a) \div (-b) = +(a \div b)\)
Example: \((-24) \div (-6) = +4\)

Rule 3: Positive ÷ Negative = Negative
Formula: \((+a) \div (-b) = -(a \div b)\)
Example: \((+24) \div (-6) = -4\)

Rule 4: Negative ÷ Positive = Negative
Formula: \((-a) \div (+b) = -(a \div b)\)
Example: \((-24) \div (+6) = -4\)

Step 1: Find the absolute values (ignore signs)

Step 2: Divide the absolute values

Step 3: Apply the sign rule to determine final sign

Parentheses / Brackets

Exponents / Orders

Multiplication and Division (left to right)

Addition and Subtraction (left to right)

Step 1: Solve inside parentheses first

Step 2: Calculate exponents/powers

Step 3: Perform multiplication and division from left to right

Step 4: Perform addition and subtraction from left to right

Step 5: Apply integer operation rules at each step

Example 1: \(-8 + 3 \times 4\)
Step 1: Multiply first → \(3 \times 4 = 12\)
Step 2: Add → \(-8 + 12 = 4\)
Answer: \(4\)

Example 2: \((5 - 9) \times (-2)\)
Step 1: Parentheses → \(5 - 9 = -4\)
Step 2: Multiply → \((-4) \times (-2) = +8\)
Answer: \(8\)

Example 3: \(-6 \div 2 + 4 \times (-3)\)
Step 1: Divide → \(-6 \div 2 = -3\)
Step 2: Multiply → \(4 \times (-3) = -12\)
Step 3: Add → \(-3 + (-12) = -15\)
Answer: \(-15\)

⚡ Remember: The number line is your best friend! Integers to the right are always greater. ⚡