Problem 78 - Prealgebra

Question

Without pencil and paper or a calculator. Is \(-20\) closer to \(-40\) or \(10 ?\)

Step-by-Step Solution

Answer

(-20) is closer to (-40) than 10.

Step 1: Understanding Differences

First, let's consider the number line and find how far (-20) is from each number. For (-40), the distance is |(-20) - (-40)| = |20| = 20. For 10, the distance is |(-20) - 10| = |-30| = 30.

Step 2: Compare Distances

Now that we have both distances: 20 is the distance between (-20) and (-40), and 30 is the distance between (-20) and 10. Compare these distances to determine which is smaller.

Step 3: Conclusion

Since 20 < 30, (-20) is closer to (-40) than it is to 10.

Key Concepts

Number Line

Visualizing integers on a number line helps us compare their positions easily. The number line is a simple horizontal line where each point represents a number.
The left side of the line holds negative numbers, and the right side holds positive numbers. Zero is placed in the middle.
This visual tool can assist us in quickly identifying how far apart numbers are from each other. When we talk about the number line, think of standing at a certain point and figuring out your distance to other numbers on this line. For instance, if we stand at \(-20\), we want to see how many steps we need to take to reach \(-40\) and \ 10\.
Understanding this helps not only with comparing distances but also with solidifying the concept of numerical positioning.

Absolute Value

Absolute value is a concept that simplifies how we read and understand the distance between numbers. It refers to the distance of a number from zero without regard to direction (positive or negative).
For instance, both \(-20\) and \ 20\ have an absolute value of \ 20\ because they are 20 units away from zero.

• The symbol for absolute value is vertical bars: \ |x| \ means the absolute value of \ x\.
• Whether a number is negative or positive, its absolute value is always non-negative.

Using absolute value, we solve questions like whether \(-20\) is closer to \(-40\) or \ 10\ without worrying about positive or negative signs – only distance matters.

Distance Comparison

Once we understand the number line and absolute values, comparing distances becomes straightforward. To compare distances, we are interested in finding out which numerical difference is smaller. For the specific exercise of finding if \(-20\) is closer to \(-40\) or \ 10\, we calculate:

• the distance between \(-20\) and \(-40\) is \ |(-20) - (-40)| = |20| = 20\,
• the distance between \(-20\) and \ 10\ is \ |-30| = 30\.

We find the smallest distance from these calculations. Since \ 20 < 30\, the result is that \(-20\) is closer to \(-40\) than it is to \ 10\.