Pre-Algebra
Table of Contents
Unit 3 Overview: Integers
3.1 Introduction to Integers
3.2 Add Integers
3.3 Subtract Integers
3.4 Multiply and Divide Integers
3.5 Solve Equations Using Integers; The Division Property of Equality

➕pre-algebra review

3.3 Subtract Integers

Citation:

Subtracting integers can be tricky, but it's a crucial skill in math. We'll explore methods like using number lines and counters to visualize subtraction, and learn how to simplify expressions by converting subtraction to addition of opposites.

We'll also dive into real-world applications, like temperature changes and financial transactions. By mastering these concepts, you'll be better equipped to handle more complex math problems and everyday calculations involving negative numbers.

Subtraction of Integers

Subtraction with number lines

  • Number line method involves starting at the minuend (first number) on the number line
    • Move left if the subtrahend (second number) is positive
    • Move right if the subtrahend is negative
    • The endpoint represents the difference (result)
  • Counter method uses two colors of counters to represent positive and negative integers
    • Start with counters equal to the absolute value of the minuend (positive counters for positive minuend, negative counters for negative minuend)
    • Remove positive counters if the subtrahend is positive
    • Add negative counters if the subtrahend is negative
    • Remaining counters represent the difference

Simplification of integer expressions

  • Subtracting a positive integer is equivalent to adding its opposite (negative)
    • $a - b = a + (-b)$ (subtracting 5 is the same as adding -5)
  • Subtracting a negative integer is equivalent to adding its opposite (positive)
    • $a - (-b) = a + b$ (subtracting -3 is the same as adding 3)
  • Subtract integers with the same sign by finding the absolute difference and using the sign of the larger absolute value
    • $-a - (-b) = -a + b = b - a$ when $|b| > |a|$ (-7 - (-3) = -7 + 3 = 3 - 7 = -4)
    • $a - b$ when $|a| > |b|$ (8 - 5 = 3)
  • Subtract integers with different signs by adding their absolute values and using the sign of the larger absolute value
    • $a - (-b) = a + b$ (4 - (-6) = 4 + 6 = 10)
    • $-a - b = -(a + b)$ (-9 - 2 = -(9 + 2) = -11)

Evaluation of variable expressions (algebraic expressions)

  • Substitute given values for variables in the expression
  • Follow order of operations (PEMDAS)
    1. Parentheses
    2. Exponents
    3. Multiplication and Division (left to right)
    4. Addition and Subtraction (left to right)
  • Simplify the expression using rules for subtracting integers
    • If $x = 3$ and $y = -5$, evaluate $2x - 3y$
      1. $2(3) - 3(-5)$
      2. $6 - (-15)$
      3. $6 + 15 = 21$

Word problems to algebraic expressions

  • Identify unknown quantity and assign a variable
  • Translate word problem into algebraic expression
    • Positive integers for quantities added or gained
    • Negative integers for quantities subtracted or lost
  • Solve algebraic expression to find value of unknown quantity
    • "John has $10. He spends $7 on lunch and earns $5 for helping a friend. How much money does he have now?"
      • Let $x$ represent John's final amount
      • $x = 10 - 7 + 5$
      • $x = 8$, so John has $8

Real-world applications of subtraction

  • Temperature changes
    • Subtracting positive temperature change decreases temperature (subtracting 5°C means temperature drops by 5°C)
    • Subtracting negative temperature change increases temperature (subtracting -3°C means temperature rises by 3°C)
  • Elevation changes
    • Subtracting positive elevation change decreases elevation (moving downward)
    • Subtracting negative elevation change increases elevation (moving upward)
  • Financial transactions
    • Subtracting positive amount decreases balance (withdrawal or expense)
    • Subtracting negative amount increases balance (deposit or income)

Key Concepts in Integer Subtraction

  • Integer: A whole number that can be positive, negative, or zero
  • Additive inverse: Two numbers that sum to zero (e.g., 5 and -5)
  • Inverse operation: Subtraction is the inverse operation of addition
    • This relationship is used to solve equations involving subtraction

Key Terms to Review (23)

Number Line: A number line is a visual representation of the number system, where numbers are arranged in a linear fashion along a horizontal or vertical axis. It serves as a fundamental tool in understanding and working with various numerical concepts, including whole numbers, integers, fractions, and rational and irrational numbers.
Commutative Property: The commutative property is a fundamental mathematical principle that states the order in which two numbers are combined does not affect the final result. This property applies to addition and multiplication operations, allowing the terms to be rearranged without changing the outcome.
Addition: Addition is a fundamental mathematical operation that combines two or more numbers or quantities to find their sum. It is a way of combining values to obtain a total or overall amount. This key term is essential in understanding various mathematical concepts and operations within the context of pre-algebra.
Associative Property: The associative property is a fundamental mathematical principle that allows the grouping of numbers or variables in an expression to be changed without affecting the overall result. It is a key concept that applies to various arithmetic operations, including addition, multiplication, and even certain algebraic expressions.
Subtrahend: The subtrahend is the number that is being subtracted from another number in a subtraction operation. It represents the quantity that is being taken away or removed from the minuend, which is the number being subtracted from.
Minuend: The minuend is the number from which another number is to be subtracted. It is the larger of the two numbers involved in a subtraction operation, and it represents the original quantity or amount before the subtraction takes place.
Difference: The difference between two numbers is the amount by which one number exceeds the other. It is the result of subtracting one number from another and represents the magnitude of the separation between the two values.
Inverse Operation: An inverse operation is a mathematical operation that undoes or reverses the effect of another operation. It is a fundamental concept in mathematics that allows for the solving of equations and the manipulation of numerical expressions.
Algebraic Expression: An algebraic expression is a mathematical phrase that combines variables, numbers, and operations to represent a value or relationship. It is a fundamental concept in algebra that allows for the representation and manipulation of quantities without specific numerical values.
Order of Operations: The order of operations is a set of rules that dictate the sequence in which mathematical operations should be performed to evaluate an expression. This term is crucial in the context of evaluating, simplifying, and translating expressions, as well as solving equations using various properties of equality.
Absolute Value: Absolute value is a mathematical concept that represents the distance of a number from zero on the number line, regardless of the number's sign. It is a way to describe the magnitude or size of a number without considering its positive or negative direction.
Additive Inverse: The additive inverse of a number is the opposite value that, when added to the original number, results in a sum of zero. It is a fundamental concept in the context of integers and other number systems.
Minus Sign: The minus sign is a mathematical symbol used to indicate subtraction or a negative value. It represents the operation of taking one number away from another, or denoting that a number is less than zero.
Negative Sign: The negative sign is a mathematical symbol (-) that indicates the opposite or inverse of a positive quantity. It is used to represent negative numbers, subtraction, and certain algebraic operations that result in a decrease or a value less than zero.
Opposite: The opposite of a number or quantity is the value that is equal in magnitude but opposite in direction or sign. Opposites represent the inverse or reverse of a given entity, concept, or relationship.
Zero Pair: A zero pair is a set of two integers, one positive and one negative, that add up to zero. It is a fundamental concept in the context of subtracting integers, as zero pairs can be used to manipulate expressions and simplify calculations.
Keep-Change-Change Rule: The keep-change-change rule is a method used to subtract integers, where the sign of the second number is changed, and the absolute values of the numbers are then subtracted. This rule helps simplify the process of subtracting integers and ensures consistent and accurate results.
Adding the Opposite: Adding the opposite, also known as additive inverse, is a mathematical operation where a number is added to its negative counterpart, resulting in a sum of zero. This concept is particularly relevant in the context of subtracting integers, as it allows for the simplification of subtraction problems by converting them into addition problems.
Negative Integer: A negative integer is a whole number that is less than zero, representing a quantity that is the opposite or inverse of a positive number. Negative integers are commonly used in the context of subtracting integers, where they play a crucial role in determining the direction and magnitude of the difference between two values.
Positive Integer: A positive integer is a whole number greater than zero. It is a fundamental concept in mathematics that represents quantities, amounts, or positions in a numerical sequence. Positive integers are essential in the context of subtracting integers, as they provide a clear understanding of the direction and magnitude of the operation.
Chip Model: The chip model is a visual representation used to understand the process of subtracting integers. It involves the use of physical or virtual chips, where each chip represents a unit, to demonstrate the subtraction of positive and negative numbers.
Two-Color Counter: A two-color counter is a physical device used to keep track of the relative position of two different integers on a number line. It is commonly employed in the context of subtracting integers, where the counter helps visualize the process of moving left or right on the number line to find the difference between two numbers.
Integer: An integer is a whole number, either positive, negative, or zero. Integers are fundamental mathematical concepts that play a crucial role in various areas of mathematics, including arithmetic, algebra, and number theory.