Problem 54 - Elementary Algebra

Question

For the following problems, on the number line, how many units (intervals) are there bet ween? -8 and \(0 ?\)

Step-by-Step Solution

Answer

Answer: There are 8 units (intervals) between -8 and 0 on the number line.

Step 1: Understanding the Problem

We are given two numbers on the number line, -8 and 0, and we need to count how many units (intervals) are between them.

Step 2: Find the difference between the numbers

To find the difference between -8 and 0, we will subtract -8 from 0: \(0 - (-8)\). Recall that subtracting a negative number is the same as adding its absolute value, so we have: \(0 + 8\).

Step 3: Calculate the difference

Now we will calculate the sum from the previous step: \(0 + 8 = 8\).

Step 4: Interpret the result

The difference between the two numbers is 8, which means there are 8 units (intervals) between -8 and 0 on the number line.

Key Concepts

Absolute Value

Understanding absolute value is crucial for solving problems like finding the distance between numbers on a number line. Absolute value refers to how far a number is from zero, regardless of its direction. It's represented as \(|x|\) for a number \(x\). For example, the absolute value of both -8 and 8 is 8. This is because they are 8 units away from zero, just in opposite directions. When working with number lines, knowing that substracting a negative number is like adding a positive, helps make calculations straightforward.

Integers

Integers are a basic type of number, including all whole numbers and their negative counterparts. These include numbers like -3, 0, and 7. When dealing with number lines, integers are important because they provide a simple way to mark and count intervals. On a number line, negative integers appear to the left of zero, while positive integers are to the right.

• Negative integers, like -8, are less than zero.
• Positive integers, like 8, are greater than zero.
• Zero itself is an integer that is central on the number line.

Understanding integers helps in visualizing distances and differences between numbers much like we see in the exercise.

Subtraction

Subtraction is more than just taking numbers away; it's about finding the difference between numbers. When we are asked to find how many intervals exist between two numbers on a number line, subtraction is often the underlying operation used. In the example of finding the distance between -8 and 0, we subtract -8 from 0. The operation looks like this: \(0 - (-8)\).

Subtracting a negative number can initially seem confusing. However, by turning the subtraction of a negative into an addition, as \(0 - (-8) = 0 + 8\), it becomes clearer. This is because subtracting a negative number uses the character of absolute value, which helps provide a positive result. The subtraction process thus tells us there are 8 intervals between the two numbers.