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Chapter 1: Problem 121

Find the distance between the given numbers on a number line. 78 and -15

Short Answer

Expert verified
The distance is 93.

Step by step solution

01

Understanding the Problem

We are asked to find the distance between two numbers on a number line: 78 and -15. This means finding the absolute difference between these numbers.
02

Distance Formula on a Number Line

To find the distance between two points on a number line, we use the formula: \[ \text{Distance} = |a - b| \text{ or } |b - a| \]where \(a\) and \(b\) are the numbers given.
03

Substitute the Values

Substitute 78 for \(a\) and -15 for \(b\) into the distance formula: \[ \text{Distance} = |78 - (-15)| \]
04

Simplify the Expression

Simplify the expression inside the absolute value bars:\[ 78 - (-15) = 78 + 15 = 93 \]
05

Calculate the Absolute Value

Find the absolute value of the simplified expression. Since the absolute value of 93 is 93, we have:\[ \text{Distance} = |93| = 93 \]
06

Conclusion

The distance between 78 and -15 on a number line is 93.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value
When talking about absolute value, think of it as the distance of a number from zero on a number line, regardless of direction. This means the absolute value of a number is always non-negative.
  • Symbol: We use vertical bars to denote absolute value, like this: \(|a|\).
  • Basic Example: For the number 5, the absolute value is \(|5| = 5\). For the number -5, it's also \(|-5| = 5\).
  • Real-World Analogy: Imagine you're talking about the actual miles you walk to get somewhere, regardless of the direction you take. That's similar to absolute value.
Whenever you're calculating the actual distance between two points, like in our exercise, absolute value ensures we're only interested in the magnitude, ignoring any negative sign.
Number Line
A number line is a straight line where each point corresponds to a real number. It's a powerful visual tool that helps us understand different numbers and their relationships.
  • Zero at the Center: Typically, zero is in the center, with positive numbers to the right and negative numbers to the left.
  • Spacing: Consistent intervals allow us to see the distance and values clearly. Each tick mark represents an equal magnitude away from neighboring numbers.
  • Understanding Numbers: Use the number line to grasp how far apart numbers are, like 78 and -15 in our example. It clearly shows why their distance is calculated using absolute value.
For the given numbers on our number line, finding the distance between 78 and -15 helps visualize how far these numbers are from each other, reinforcing the idea of absolute distance.
Distance Formula
To determine the distance between two points on a number line, we use a simple yet efficient formula. This is crucial in various fields, including math and physics.
  • Formula: The distance is calculated as \[ \text{Distance} = |a - b| \,\]/\[ \text{or} \ |b - a| \.\]
  • Substitute Values: In exercises, substitute the given values into the formula for calculation. For example, with numbers 78 and -15, the formula becomes \[\text{Distance} = |78 - (-15)|.\]
  • Simplification: Simplify inside the absolute value: \(|78 + 15| = 93\). Since 93 is already positive, the absolute value remains 93.
  • Conclusion: The distance on the number line between these two points is \(|93| = 93\).
This formula highlights the practical use of absolute value and ensures the result is a non-negative distance, consistent with our intuitive understanding of how far apart numbers are.

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