Problem 7 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated subtraction. $$8-14$$

Step-by-Step Solution

Answer

The result of the subtraction \(8 - 14\) is \(-6\).

Step 1: Perform the Subtraction

The given subtraction is \(8-14\). Since 14 is larger than 8, the result will be negative.

Step 2: Calculate the_difference

To calculate the difference, subtract 8 from 14, which gives 6. Since the larger number is being subtracted from the smaller one, the result will be -6.

Key Concepts

Negative Numbers

Negative numbers extend the number line below zero to include numbers less than zero. When you perform operations such as subtraction, you might encounter them.
Understanding negative numbers involves grasping that they represent values less than zero. Imagine a thermometer: the lower the temperature goes, the more negative the number becomes.

• **Positive and Negative Symbols**: These symbols indicate the sign of a number; a negative number has a minus sign (-) while a positive number does not.
• **On the Number Line**: Numbers to the left of zero are negative, and as they move right towards zero, they increase in value.
• **Operation with Negatives**: Negative values affect operations such as subtraction, often resulting in another negative number.

Recognizing negative numbers is important in algebra and everyday math because it affects how we measure or predict outcomes.

Subtraction Techniques

Subtraction involves finding the difference between two numbers and can result in a positive or negative outcome depending on their values. The process of subtraction can be made easier by using techniques.

• **Reverse Thinking**: You can think of subtraction as "adding the opposite." For instance, subtracting 14 from 8 can be thought of as adding -14 to 8.
• **Ordering the Numbers**: It’s helpful to decide which number is larger, which directly affects whether your result is positive or negative.
• **Grouping Method**: Sometimes it's easier to first subtract in parts. Break down complicated numbers into smaller, manageable parts and subtract those.

These techniques help simplify the subtraction process, especially when dealing with larger or negative numbers.

Basic Algebraic Operations

Within algebra, basic operations include addition, subtraction, multiplication, and division. Grasping these is essential for solving equations and expressions.
Algebraic operations follow an order of precedence known as "order of operations."

• **Addition and Subtraction**: These are inverse operations, and understanding their relationship helps in solving algebraic equations.
• **Multiplication and Division**: Similarly, these are opposite operations and have a stronger precedence over addition and subtraction.
• **Commutative and Associative Properties**: Addition and multiplication are commutative and associative--meaning it doesn’t matter how you order or group them, the result is the same.

Knowing these operations and properties forms the basis of solving more complex algebraic problems and aids in understanding relationships within equations.