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Chapter 2: Problem 16

Which number is closer to 2 on the number line? \(-5\) or 8

Short Answer

Expert verified
8 is closer to 2 than -5.

Step by step solution

01

Identify the Given Numbers and Target

The given numbers are -5 and 8. The target number is 2. The goal is to determine which of the given numbers is closer to 2 on the number line.
02

Calculate the Distance to 2

To find the distance between each number and 2, subtract 2 from each number and then take the absolute value. This is because distance on a number line is always a positive value. Let's start with -5:data: Distance = |-5 - 2| = |-7| = 7Now for 8:Distance = |8 - 2| = |6| = 6
03

Compare the Distances

The distance from -5 to 2 is 7 units, and the distance from 8 to 2 is 6 units. Since 6 units is less than 7 units, 8 is closer to 2 than -5.

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Key Concepts

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

absolute value
The concept of absolute value is crucial in understanding distances on a number line. Absolute value refers to the distance a number is from zero on the number line, regardless of direction. This means that whether the number is positive or negative, the absolute value is always positive.
  • For example, the absolute value of -5 is 5, written as \(|-5| = 5\).
  • Similarly, the absolute value of 8 is 8, written as \(|8| = 8\).
In our exercise, we use absolute value to find out how far the given numbers (-5 and 8) are from the number 2. By doing so, we convert any negative values we encounter into their positive counterparts, making it easier to directly compare distances.
distance calculation
Calculating distance on a number line involves using absolute value to ensure that all distances are positive. Here's how we break it down:
First, subtract the target number from each given number. After that, take the absolute value of the result to get the distance.
For instance:
  • Distance from -5 to 2: Subtract 2 from -5 to get -7, and then take the absolute value to get \(|-7| = 7\).
  • Distance from 8 to 2: Subtract 2 from 8 to get 6, and then take the absolute value to get \(|6| = 6\).
The process ensures all distances are positive, making it simple to compare how far each number is from the target number.
comparing distances
Once the distances are calculated, the next step is to compare them to determine which is shorter. In our exercise:
  • The distance from -5 to 2 is 7 units.
  • The distance from 8 to 2 is 6 units.
Since 6 units is less than 7 units, it tells us that 8 is closer to 2 than -5. Comparing distances this way on a number line helps in making clear decisions about proximity. It's a method that can be extended to any set of numbers and target values on a number line, making it a versatile tool in math.

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