Part 1: Core Concepts & Rules (Integer Operations & Absolute Value)

Definition of Absolute Value:
- Absolute value ( $\text{AV}$ ) is defined as the distance a number is from zero ( $0$ ) on the number line.
- Because distance is a geometric measure and cannot be negative, absolute value is always positive (or zero).
- Symbolic representation: Absolute value is designated by two vertical bars around a value, written as $|x|$ .
Illustrative Example & Visual Interpretation:
- Example: $|-5| = 5$
- Explanation: The distance from $-5$ to $0$ is exactly $5$ units.
- Geometric visualization: If you locate $-5$ on a horizontal number line and draw a directed arc toward the origin at $0$ , the total path spans a magnitude of $5$ units, confirming that $|-5| = 5$ .

Rule for Adding Integers with the Same Sign:
- Procedure: Add the absolute values of the numbers together and keep the shared, original sign.
- Example:
  - $(-4) + (-3) = -7$
  - Step-by-step breakdown: Both numbers have negative signs. Combine their numerical values ( $4 + 3 = 7$ ) and keep the negative sign, yielding $-7$ .
Rule for Adding Integers with Different Signs:
- Procedure: Subtract the smaller absolute value from the larger absolute value, then keep the sign of the number with the larger absolute value ( $\text{AV}$ ).
- Example:
  - $(-5) + 3 = -2$
  - Step-by-step breakdown: The absolute values are $|-5| = 5$ and $|3| = 3$ . Subtract the smaller from the larger ( $5 - 3 = 2$ ). Because $-5$ has the larger absolute value ( $5 > 3$ ), apply its negative sign to the result, producing $-2$ .

Rule for Subtracting Integers:
- Procedure: Keep the first number, change the subtraction sign ( $-$ ) to an addition sign ( $+$ ), and change the sign of the last number to its opposite sign.
- This procedure is commonly referred to as the Keep-Change-Change method:
  - Keep: Retain the first integer exactly as written.
  - Change: Change the subtraction operator ( $-$ ) to addition ( $+$ ).
  - Change: Change the second integer to its additive inverse (opposite sign).
Worked Examples:
- Example 1 (Subtracting a negative number):
  - $5 - (-3) = 5 + 3 = 8$
  - Step-by-step breakdown: Keep the first number ( $5$ ), change the subtraction operation to addition ( $+$ ), and change the negative three ( $-3$ ) to positive three ( $3$ ). Then evaluate $5 + 3 = 8$ .
- Example 2 (Subtracting a positive number from a negative number):
  - $-6 - 4 = -6 + (-4) = -10$
  - Step-by-step breakdown: Keep the first number ( $-6$ ), change the subtraction operation to addition ( $+$ ), and change the positive four ( $4$ ) to negative four ( $-4$ ). Finally, apply the same-sign addition rule: $6 + 4 = 10$ and retain the negative sign, resulting in $-10$ .

Multiplication Sign Rules:
- Same Signs: Multiplying two integers with the same sign (both positive or both negative) always results in a positive product ( $(+)$ ).
- Different Signs: Multiplying two integers with different signs (one positive and one negative) always results in a negative product ( $(-)$ ).
Worked Examples:
- Same Sign Example:
  - $(-4) \times (-3) = 12$
  - Explanation: Both factors have identical negative signs; therefore, their product is positive ( $+12$ ).
- Different Sign Example:
  - $(-5) \times 3 = -15$
  - Explanation: The two factors have differing signs (negative and positive); therefore, the product is negative ( $-15$ ).

Division Sign Rules:
- Same Signs: Dividing two integers with the same sign (both positive or both negative) results in a positive quotient ( $(+)$ ).
- Different Signs: Dividing two integers with different signs (one positive and one negative) results in a negative quotient ( $(-)$ ).
Worked Examples:
- Same Sign Example:
  - $(-20) \div (-4) = 5$
  - Explanation: The dividend ( $-20$ ) and the divisor ( $-4$ ) share the same negative sign, producing a positive quotient ( $5$ ).
- Different Sign Example:
  - $18 \div (-3) = -6$
  - Explanation: The dividend ( $18$ ) and divisor ( $-3$ ) have opposite signs (positive divided by negative), resulting in a negative quotient ( $-6$ ).