Problem 33 - Algebra 1

Question

Find the opposite of the number. $$8$$

Step-by-Step Solution

Answer

The opposite of 8 is -8.

Step 1: Identify the given number

The number given in the exercise is 8.

Step 2: Apply the operation to find the opposite of the number

The opposite of a number is found by multiplying the number by -1. Therefore, do the operation \(8 \times -1\).

Step 3: Write down the result

After performing the operation, the result is -8.

Key Concepts

Integers and Opposite Values

When we talk about integers, we're referring to whole numbers and their negatives. This includes numbers like -3, 0, and 4. Each integer has an opposite, also known as its additive inverse. The opposite of an integer is the number that, when added to the original number, equals zero. For example, the opposite of 3 is -3, because when you add them together (3 + -3), their sum is 0.

To find the opposite of any integer, you simply change its sign. If the original integer is positive, its opposite is negative. If it's negative, its opposite will be positive. So in exercise terms, considering the integer 8, its opposite is -8. This concept is a fundamental part of number theory and is frequently used in various algebraic operations.

Multiplying by Negative One

The mathematical operation that changes a number to its opposite is multiplication by -1. Why does this work? Essentially, when you multiply any number by 1, its value doesn't change because 1 is the multiplicative identity. However, introducing the negative sign inverts the number's sign.

If you're multiplying a positive number by -1, the product is the same number with a negative sign, making it the opposite. Conversely, if you start with a negative and multiply by -1, the double negative turns positive. Following our example where the integer is 8, when we compute \(8 \times -1\), we get -8. It's crucial for students to understand this operation as it is a building block for handling more complex algebraic expressions.

Basic Algebraic Operations

At the heart of algebra are basic operations which include addition, subtraction, multiplication, and division. These operations make up the foundation for more advanced concepts in mathematics. An understanding of how to work with integers — particularly how to find and use opposite values — is essential in algebra.

When an algebraic equation requires you to cancel out terms, you'll find opposites quite handy. If you have \(x + 8 = 0\), solving for \(x\) means you need to add the opposite of 8 to both sides. This basic understanding of opposites also sets the stage for solving inequalities, working with absolute values, and understanding the coordinate plane where each point has an opposite. It's about recognizing patterns and relationships between numbers, which is the core of algebra.