Problem 52 - Elementary Algebra

Question

Find the distance between the given numbers on a number line. $$ -100 \text { and }-130 $$

Step-by-Step Solution

Answer

The distance is 30 units.

Step 1: Identify the Two Points

The points given on the number line are -100 and -130. These numbers represent their own positions.

Step 2: Calculate the Distance

To find the distance between two points on a number line, subtract the smaller number (-130) from the larger number (-100). This will give us the absolute value of the distance between the two points:\[ |-100 - (-130)| = |-100 + 130| = |30| = 30 \]

Step 3: Understand the Absolute Value

The concept of absolute value is used here to ensure the result is a non-negative number representing the distance between the numbers, regardless of their positions on the number line.

Key Concepts

Distance Calculation

Calculating the distance on a number line is like finding out how far apart two points are, without worrying about their direction. Imagine you have a number line—this line is your trusty tool for picturing numbers in order.
It helps us see where each number sits compared to another. Here's what you need to know:

• Numbers get larger as you move to the right on the line.
• Numbers get smaller as you move to the left.
• The distance is simply the amount of space between two points.

To calculate this, you start by identifying the two numbers you're working with—like -100 and -130.
Then, you perform simple subtraction and take the absolute value of the result.
This gives you the 'distance', a positive number that shows how many units of space lie between the two chosen points.

Absolute Value

Absolute value is a key idea when it comes to distance. It allows us to focus on the size of a number (or result), not its direction.
When you calculate the distance between two points on a number line, the absolute value ensures you end up with a positive number.
Here's a simple way to think about it:

• The absolute value of a number is its distance from zero, without considering if it's positive or negative.
• In symbols, \(|a|\) means the absolute value of 'a'.
• So, \(|-5|\) becomes 5, just like \(|5|\) also remains 5.

This is handy, because distance should never be a negative number—it just represents how much 'space' sits between two points, nevermind the placement on the number line.

Integer Subtraction

Subtracting integers is like figuring out how much of one number you have "compared" to another. It shows the difference between them.
When numbers are negative, this concept can seem a bit tricky.Here's how you navigate integer subtraction:

• First, identify which number is larger or smaller in terms of position (not value).
• Then, subtract the smaller from the larger if both numbers are negative.
• The equation becomes subtraction of negatives, which flips into addition due to double negatives.
• Example: \(-100 - (-130)\) turns into \(-100 + 130 = 30\).

This process ensures you're comparing the right parts of numbers for a clear understanding of how different or close they truly are.
Using subtraction and absolute value, you ensure you measure pure distance on a number line. This helps clear up confusion about negative placements!