3.4 Multiply and Divide Integers

Multiplying and dividing integers can be tricky, but it's crucial for solving real-world problems. The key is understanding how positive and negative numbers interact. When multiplying or dividing, same signs give a positive result, while different signs yield a negative one.

These operations are essential for simplifying algebraic expressions and solving equations. By mastering integer operations, you'll be better equipped to tackle more complex math problems and apply these skills to everyday situations involving temperature changes, finances, and more.

Multiplying and Dividing Integers

Multiplication of positive and negative integers

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• Multiplying two integers with the same sign yields a positive result
  • Positive × Positive = Positive (2 × 3 = 6)
  • Negative × Negative = Positive (-4 × -5 = 20)
• Multiplying two integers with different signs yields a negative result
  • Positive × Negative = Negative (3 × -7 = -21)
  • Negative × Positive = Negative (-6 × 4 = -24)
• The product's absolute value equals the product of the factors' absolute values
  • $|a × b| = |a| × |b|$ ($|-6 × 4| = |-6| × |4| = 24$)

Division operations with integers

• Dividing two integers with the same sign yields a positive result
  • Positive ÷ Positive = Positive (12 ÷ 3 = 4)
  • Negative ÷ Negative = Positive (-20 ÷ -5 = 4)
• Dividing two integers with different signs yields a negative result
  • Positive ÷ Negative = Negative (18 ÷ -6 = -3)
  • Negative ÷ Positive = Negative (-24 ÷ 8 = -3)
• The quotient's absolute value equals the quotient of the dividend and divisor's absolute values
  • $|a ÷ b| = |a| ÷ |b|$ ($|-24 ÷ 8| = |-24| ÷ |8| = 3$)
• Division by zero is undefined for all integers (10 ÷ 0 is undefined)

Properties of Integer Operations

• Identity property: Multiplying any integer by 1 or dividing it by 1 results in the same integer
• Opposite integers: Two integers with the same absolute value but different signs (e.g., 5 and -5)
• Additive inverse: The opposite of an integer, which when added to the original integer, results in zero
• Multiplicative inverse: The reciprocal of a non-zero integer, which when multiplied by the original integer, results in 1

Simplification of integer algebraic expressions

• Follow the order of operations: PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right)
• Simplify expressions inside parentheses first (2 × (3 - 5) = 2 × -2 = -4)
• Perform multiplication and division operations from left to right (12 ÷ -3 × 2 = -4 × 2 = -8)
• Perform addition and subtraction operations from left to right (-6 + 4 - 3 = -2 - 3 = -5)
• Apply the rules for multiplying and dividing integers when simplifying expressions (-3 × (4 - 7) ÷ 2 = -3 × -3 ÷ 2 = 9 ÷ 2 = 4.5)

Variable expressions with integer values

• Substitute the given integer value for the variable in the expression ($3x - 2$ when $x = -4$ becomes $3(-4) - 2$)
• Simplify the expression using the order of operations and the rules for multiplying and dividing integers ($3(-4) - 2 = -12 - 2 = -14$)
• Evaluate the expression to find the solution ($-2y^2 + 5y - 3$ when $y = -1$ becomes $-2(-1)^2 + 5(-1) - 3 = -2 - 5 - 3 = -10$)
• Use the integer number line to visualize operations and relationships between integers

Word problems to algebraic expressions

• Identify the unknown quantity and represent it with a variable (let $x$ represent the number of apples)
• Translate the word problem into an algebraic expression using the variable
  • Use keywords to determine the operation:
    1. "Sum," "more than," "increased by" indicate addition (3 more than twice a number: $2x + 3$)
    2. "Difference," "less than," "decreased by" indicate subtraction (5 less than a number: $x - 5$)
    3. "Product," "times," "of" indicate multiplication (3 times a number: $3x$)
    4. "Quotient," "divided by," "ratio" indicate division (a number divided by 4: $x ÷ 4$ or $\frac{x}{4}$)
• Solve the algebraic expression using the given integer values and the rules for multiplying and dividing integers (if $x = -6$, then $2x + 3 = 2(-6) + 3 = -12 + 3 = -9$)

Applying Integer Operations

Solve real-world problems

• Read the problem carefully and identify the given information and the question being asked
• Represent the unknown quantity with a variable (let $t$ represent the temperature in °C)
• Write an algebraic expression or equation based on the problem statement (the temperature decreased by 7°C from its initial value: $t - 7$)
• Substitute the given integer values into the expression or equation (if the initial temperature was 5°C, then $5 - 7 = -2$)
• Simplify the expression or solve the equation using the rules for multiplying and dividing integers
• Interpret the solution in the context of the original problem (the final temperature is -2°C)