Problem 52 - Beginning and Intermediate Algebra

Question

Find (a) the additive inverse and (b) the absolute value. 8.1

Step-by-Step Solution

Answer

(a) -8.1, (b) 8.1

Step 1: Understand the Additive Inverse

The additive inverse of a number is what you add to the number to get zero. For a number, the additive inverse is simply the negative of that number.

Step 2: Calculate the Additive Inverse

For the number 8.1, the additive inverse is -8.1 because 8.1 + (-8.1) = 0.

Step 3: Understand Absolute Value

The absolute value of a number is the distance between that number and zero on a number line, regardless of direction. It is always a non-negative number.

Step 4: Calculate the Absolute Value

For the number 8.1, the absolute value is simply 8.1 because it is already a positive number. The absolute value of 8.1 is denoted as \(|8.1| = 8.1\).

Key Concepts

Additive Inverse

The concept of the additive inverse is pretty straightforward but crucial to understanding algebra. The additive inverse of a number is what you would add to that number to get zero. This means that for any given number, its additive inverse is its negative counterpart. For instance, the additive inverse of 3 is -3, because 3 + (-3) equals 0.

Let's break this down further:

• If you have a positive number, like 8.1, its additive inverse will be -8.1.
• If you have a negative number, for example, -8.1, its additive inverse will be 8.1.

Knowing this, we can easily find the additive inverse of any number. Remember, the goal is to find what needs to be added to make the sum zero. So, 8.1 and -8.1 are additive inverses of each other because 8.1 + (-8.1) = 0.

Absolute Value

When we talk about absolute value, we're talking about how far a number is from zero, without considering the direction. It is a measure of the magnitude of a number on the number line. The absolute value is always non-negative. For instance, the absolute value of both 5 and -5 is 5, since both are five units away from zero.

To better understand:

• The absolute value of 8.1 is simply 8.1 because it is already positive.
• The absolute value of -8.1 is also 8.1, because it’s 8.1 units away from zero.

We denote the absolute value using vertical bars. For instance, the absolute value of 8.1 is written as \(|8.1| = 8.1\) and the absolute value of -8.1 is written as \(|-8.1| = 8.1\). This makes it clear that absolute value is all about distance, not direction.

Understanding Number Properties

To excel in algebra, understanding number properties is fundamental. Two of these properties are additive inverses and absolute values. Here's a clear way to think about them:

Additive Inverse:

• Each number has an additive inverse that makes the sum zero.
• It's like finding a partner in dance - they need to balance each other out to create harmony.

Absolute Value:

• The absolute value tells us how far a number is from zero, ignoring the direction.
• It’s always a positive distance.

Combining these properties helps us solve equations and understand the behaviors of numbers better. When we master these basic concepts, we can tackle more complex algebraic problems with confidence.