Problem 16 - Elementary Algebra

Question

Determine each of the values, \(-|8|\)

Step-by-Step Solution

Answer

Answer: -8

Step 1: Find the absolute value of 8

The absolute value of a number represents its distance from zero. In this case, the number is 8, which is already a positive number. So, the absolute value of 8 is equal to 8: \(|8| = 8\)

Step 2: Multiply the absolute value by -1

Now that we have found the absolute value of 8, we need to find the value of \(-|8|\). To do this, we simply multiply the absolute value (8) by -1: \(-|8| = -8\) So, the value of \(-|8|\) is -8.

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They represent values below zero and are typically used to show losses, debts, or temperatures below freezing. On the number line, you will find negative numbers to the left of zero. For instance, -3 is considered a negative number as it lies three units to the left of zero.

When dealing with negative numbers, it's important to remember that they have unique properties:

• Adding a negative number is the same as subtracting its positive counterpart.
• Subtracting a negative number is the same as adding the positive version of that number. In essence, two negatives cancel out, becoming positive.
• Negative numbers flipped in sign become positive, emphasizing their position relative to zero.

Multiplication by -1

Multiplication by -1 is a fascinating operation that effectively changes the sign of a number. If you take a positive number and multiply it by -1, you get its negative counterpart. Similarly, multiplying a negative number by -1 results in a positive version of that number. For example, if you have a number 5, multiplying by -1 gives you -5. Conversely, if you start with -5 and multiply by -1, you return to 5.

The operation can be thought of as a reflection across zero or the origin on a number line, flipping the number to the opposite side. It's a simple yet powerful operation, especially when used in solving equations or transforming functions.

Distance from Zero

The concept of distance from zero is fundamental to understanding absolute values. The absolute value of a number is the measure of how far it is from zero on a number line, disregarding its sign. This is why the absolute value of both -8 and 8 is 8, as both are eight units away from zero.

Understanding distance from zero helps in visualizing and solving equations without focusing on whether the number is positive or negative. It brings out the notion of magnitude without direction, making calculations involving absolute values simpler and more intuitive. Also, the idea of distance helps in real-world applications, such as determining the net change in elevation or calculating total distance traveled.