Expressions and Integers Lesson: Key Concepts and Examples

Lesson Overview

• What Are Integers?
• What Are Expressions?
• Working With Integers in Expressions
• Solving Expressions With Integers
• Examples of Integers and Expressions
• Practice Problems

In mathematics, expressions and integers are essential concepts that form the basis of many mathematical operations and problem-solving techniques. An integer is any whole number, whether positive, negative, or zero, that does not have any fractional or decimal component. An expression is a combination of numbers, variables, and mathematical operations (such as addition, subtraction, multiplication, and division) that represent a specific value.

This lesson will help you understand the concept of integers, how they are used in mathematical expressions, and how to solve problems involving expressions and integers. We will also explore how to simplify and evaluate expressions and how to use integers with clear and easy examples.

What Are Integers?

Integers are whole numbers that can be positive, negative, or zero. Unlike fractions or decimals, integers do not have fractional or decimal parts.

Types of Integers:

• Positive Integers: Numbers greater than zero (e.g., 1, 2, 3, 4).
  
• Negative Integers: Numbers less than zero (e.g., -1, -2, -3).
  
• Zero: Zero is neither positive nor negative, but it is an integer.
  

Examples of Integers:

• Positive Integers: 4, 10, 78
  
• Negative Integers: -3, -15, -50
  
• Zero: 0
  

Integers are used in a variety of situations, including counting, measuring, and representing financial data (such as profits and losses).

What Are Expressions?

An expression is a mathematical phrase that contains numbers, variables, and operations (addition, subtraction, multiplication, division). An expression does not have an equal sign (=), unlike an equation.

Example of an Expression:

• 3x + 5 is an expression that means "three times a number (x) plus five".
  
• 7 - 4 is an expression that means "seven minus four".
  

An expression can be evaluated by substituting specific values for the variables (if there are any) and performing the operations. For example, to evaluate 3x + 5 when x = 2, you would substitute 2 for x:

3(2)+5=6+5=11

Working With Integers in Expressions

In mathematical expressions, integers are used to represent quantities, and they are combined using arithmetic operations. When working with integers in expressions, it's important to follow the correct rules for addition, subtraction, multiplication, and division.

1. Addition of Integers:

• Same Signs: When adding two integers with the same sign (either both positive or both negative), add the absolute values and keep the common sign.
  
  • Example: 5 + 3 = 8 (both positive)
    
  • Example: -4 + (-6) = -10 (both negative)
    
• Different Signs: When adding integers with different signs, subtract the smaller absolute value from the larger absolute value and keep the sign of the integer with the larger absolute value.
  
  • Example: 5 + (-3) = 2 (positive because 5 is larger)
    
  • Example: -8 + 6 = -2 (negative because -8 is larger)
    

2. Subtraction of Integers:

To subtract integers, you can convert the subtraction into addition by adding the opposite (or add the negative of the number being subtracted):

• Example: 5 - 3 = 5 + (-3) = 2
  
• Example: -5 - (-3) = -5 + 3 = -2
  

3. Multiplication of Integers:

• Same Signs: When multiplying two integers with the same sign, the product is positive.
  
  • Example: 3 × 2 = 6
    
  • Example: -4 × -2 = 8
    
• Different Signs: When multiplying two integers with different signs, the product is negative.
  
  • Example: 3 × -2 = -6
    
  • Example: -4 × 2 = -8
    

4. Division of Integers:

• Same Signs: When dividing two integers with the same sign, the quotient is positive.
  
  • Example: 6 ÷ 2 = 3
    
  • Example: -8 ÷ -2 = 4
    
• Different Signs: When dividing two integers with different signs, the quotient is negative.
  
  • Example: 6 ÷ -2 = -3
    
  • Example: -8 ÷ 2 = -4
    

Take This Quiz:

Expressions and Integers Test! Math Trivia Quiz

Solving Expressions With Integers

Now that we've learned how to add, subtract, multiply, and divide integers, let's practice solving expressions that involve integers.

Example 1: Solving 3x - 5 when x = 2

• Substitute x = 2 into the expression:
  

3(2)−5=6−5=1

Example 2: Solving -4y + 7 when y = -3

• Substitute y = -3 into the expression:
  

−4(−3)+7=12+7=19

Example 3: Solving 5 + (-2) × 3

• Follow the order of operations (PEMDAS – Parentheses, Exponents, Multiplication, Division, Addition, Subtraction):
  

5+(−2)×3=5+(−6)=−1

Examples of Integers and Expressions

Understanding how to work with integers and expressions is not just useful for math class-it's essential for real-life situations. Here are a few examples:

1. Banking: When you deposit money into a bank account, that's a positive integer. When you withdraw money, that's a negative integer. You can use expressions to calculate your balance over time.
   
2. Temperature: In weather, negative temperatures represent cold weather, and positive temperatures represent warm weather. You can use integers and expressions to calculate the difference in temperatures.
   
3. Profit and Loss: In business, profits are positive integers, while losses are negative integers. Expressions can be used to calculate the net profit or loss.
   

Take This Quiz:

Quiz on all operations with integers

Practice Problems

Let's test your understanding of expressions and integers with a few practice problems:

1. What is the value of n in the equation 2n + 10 = 20?
   
2. Solve 5 + (-3) × 2 using the order of operations.
   
3. What is -7 + 3?
   
4. Simplify 3 + (-4) × 5.
   
5. What is the result of 12 ÷ -4?