Problem 61 - Algebra A Combined Function

Question

Find each additive inverse or opposite. See Examples 13 through 17. $$ -2 $$

Step-by-Step Solution

Answer

The additive inverse of \( -2 \) is \( 2 \).

Step 1: Understand the Concept of Additive Inverse

The additive inverse of a number is a number that, when added to the original number, results in zero. For any real number \( a \), its additive inverse is \( -a \).

Step 2: Identify the Given Number

The given number in this exercise is \( -2 \). We need to find its additive inverse.

Step 3: Apply the Additive Inverse Rule

To find the additive inverse of \( -2 \), change its sign. Thus, the additive inverse of \( -2 \) is \( 2 \).

Step 4: Verify the Result

Add the number \( -2 \) and its inverse \( 2 \). The operation \( -2 + 2 = 0 \) confirms that \( 2 \) is indeed the additive inverse because their sum is zero.

Key Concepts

Real Numbers

Real numbers are the broad category that includes every number we usually encounter in mathematics. This encompasses various types like whole numbers, integers, fractions, and decimals. You can think of real numbers as any point on the endless line known as the number line.

It is important to realize that both positive and negative numbers exist within this set. Whether a number is rational (like \(\frac{1}{2}\)) or irrational (like \(\sqrt{2}\)), it belongs to the real numbers. Understanding this range is essential because, when dealing with problems such as inverses, we consider every possible real number.

Opposite Sign

The concept of opposite sign is fundamental when dealing with numbers, particularly in the realm of opposites and inverses. Every real number has an opposite sign counterpart. This opposite sign is what you see when you simply flip the sign of a number.

For example, if you start with a positive number like 5, its opposite sign is -5. Conversely, for a negative number such as -2, its opposite sign would be 2. This switching forms the basis for finding an additive inverse.

In math, changing the sign of a number matters significantly because it underlies operations and functions, helping us to solve problems such as addition that results in zero.

Zero Sum

The zero sum concept in mathematics indicates a scenario where the sum of certain numbers results in zero. This is particularly relevant in topics like additive inverses.

When you have a number and its additive inverse, the sum will always be zero. For example, if you add +3 and -3, you get zero: \(3 + (-3) = 0\). Likewise, \(-2 + 2 = 0\), illustrating that zero sum is achieved when a number and its opposite sign are combined.

This is a common principle in algebra that adds clarity and simplifies many arithmetic problems. It is used in various mathematical applications to balance equations and maintain equality.

Algebra Basics

Understanding algebra basics is crucial for solving equations effectively. Algebra is the foundational tool in mathematics involving numbers and symbols to manipulate and form equations.

Key concepts include terms like variables, coefficients, and operations (addition, subtraction, multiplication, and division). Algebra helps translate real-world situations into mathematical expressions or equations, allowing us to solve them logically.

• Recognize and apply the principles of inverse operations.
• Use variables to represent numbers in equations.
• Solve equations by isolating variables.

Finding the additive inverse, as in our example, utilizes algebra basics by using the operation of adding a real number to its inverse to reach a zero sum.