1.4 Multiply and Divide Integers

Multiplying and dividing integers is crucial for mastering basic algebra. These operations follow specific rules based on the signs of the numbers involved. Understanding these rules helps simplify expressions and solve equations accurately.

Real-world applications of integer operations are everywhere. From calculating temperature changes to managing finances, these skills are essential. By practicing with word problems, you'll develop the ability to translate everyday situations into mathematical expressions and solve them confidently.

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 times positive equals positive (2 × 3 = 6)
  • Negative times negative equals positive (-4 × -5 = 20)
• Multiplying two integers with different signs yields a negative result
  • Positive times negative equals negative (3 × -2 = -6)
  • Negative times positive equals negative (-5 × 4 = -20)
• The absolute value of the product equals the product of the absolute values of the factors
  • $|a × b| = |a| × |b|$ ($|-3 × 4| = |-3| × |4| = 12$)
• Multiplication by 1 leaves the integer unchanged (multiplicative identity)

Division operations with integers

• Dividing two integers with the same sign yields a positive result
  • Positive divided by positive equals positive (10 ÷ 5 = 2)
  • Negative divided by negative equals positive (-12 ÷ -3 = 4)
• Dividing two integers with different signs yields a negative result
  • Positive divided by negative equals negative (15 ÷ -3 = -5)
  • Negative divided by positive equals negative (-20 ÷ 4 = -5)
• Division by zero is undefined for all integers
  • Any integer divided by zero results in an undefined value (8 ÷ 0 = undefined)

Properties of Integer Operations

• Commutative property: The order of factors does not affect the product (a × b = b × a)
• Associative property: Grouping of factors does not affect the product ((a × b) × c = a × (b × c))
• Distributive property: Multiplication distributes over addition or subtraction (a(b + c) = ab + ac)

Simplification of algebraic expressions

• Apply the order of operations: PEMDAS
  1. Parentheses
  2. Exponents
  3. Multiplication and Division (left to right)
  4. Addition and Subtraction (left to right)
• Use the rules for multiplying and dividing integers when simplifying expressions
  • Example: $-3(2x - 5) + 4(-2x + 3)$ simplifies to $-6x + 15 - 8x + 12 = -14x + 27$
• Combine like terms by adding or subtracting their coefficients
  • Like terms have the same variable and exponent ($3x^2$ and $-5x^2$ are like terms)

Substitution in variable expressions

• Replace variables with their corresponding integer values
  • If $x = -2$ and $y = 3$, then $2x - 3y$ becomes $2(-2) - 3(3)$
• Simplify the resulting expression using the order of operations
  • $2(-2) - 3(3) = -4 - 9 = -13$
• Apply the rules for multiplying and dividing integers when evaluating the expression
  • Example: If $a = -3$ and $b = 2$, then $-2ab$ evaluates to $-2(-3)(2) = 12$

Applying Integer Operations to Word Problems and Real-World Scenarios

Integer word problem conversion

• Identify given information and unknown quantities
  • "John has $20. He owes his friend$35. How much money does John need to pay back his friend?"
    • Given: John has $20, owes$35
    • Unknown: Amount John needs to pay back
• Assign variables to represent unknown quantities
  • Let $x$ represent the amount John needs to pay back
• Translate verbal descriptions into mathematical expressions
  • $x = 35 - 20$, since John needs to pay the difference between what he owes and what he has

Integer operations in real-world scenarios

• Recognize situations where integer operations apply
  • Temperature changes (positive for increase, negative for decrease)
    • "The temperature was 5°C in the morning and dropped by 8°C in the evening."
  • Profit and loss (positive for profit, negative for loss)
    • "A company made a profit of $10,000 in January but suffered a loss of$5,000 in February."
  • Elevation (positive for above sea level, negative for below sea level)
    • "A submarine is diving from 100 meters above sea level to 50 meters below sea level."
• Formulate equations or expressions based on given information and relationships
  • Evening temperature = Morning temperature + Temperature change
    • $T_\text{evening} = 5 + (-8) = -3$°C
  • Net profit = January profit + February profit
    • $\text{Net profit} = 10,000 + (-5,000) = 5,000$
  • Change in elevation = Final elevation - Initial elevation
    • $\text{Change in elevation} = -50 - 100 = -150$ meters
• Solve equations or evaluate expressions using integer operation rules
• Interpret results in the context of the original problem
  • The evening temperature was -3°C
  • The company's net profit was $5,000
  • The submarine dove 150 meters