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Chapter 1: Problem 2

Graph the following points and their opposites on the real number line: \(-3,-1\), and 5 .

Short Answer

Expert verified
Graph the points -5, -3, -1, 1, 3, and 5 on a number line.

Step by step solution

01

Identify the Points

We are given the points \(-3\), \(-1\), and \(5\). Our task is to also include their opposites. Recall that the opposite of a number is the same distance from zero on the number line, but in the opposite direction.
02

Determine Opposites

Let's determine the opposites of the given points. The opposite of \(-3\) is \(3\) because it is 3 units from zero in the opposite direction. Similarly, the opposite of \(-1\) is \(1\), and the opposite of \(5\) is \(-5\).
03

List All Points

Now, compile all the points we've identified: \(-3\), \(-1\), \(5\), and their opposites \(3\), \(1\), \(-5\). So, the complete set of points to graph is \(-5, -3, -1, 1, 3,\) and \(5\).
04

Plot Points on Number Line

Draw a horizontal number line and label it with key values. Plot each of these points on the number line. Make sure to place them at equal intervals for accuracy. Start with \(-5\) on the left and continue to \(5\).

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Key Concepts

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

Graphing Points
Graphing points on a real number line is a way to visually represent values. Imagine a straight line with evenly spaced marks indicating the integers. Each point represents a specific value along this line. It's helpful to visualize both negative and positive numbers, as well as zero.
When you need to graph specific points, like \(-3, -1,\) and \(+5,\), you locate them on this line based on their value relative to zero.
  • Negative numbers, like \(-3\), are found to the left of zero.
  • Positive numbers, like \(+5\), are to the right of zero.
By marking these points, you create a visual aid that makes it easier to understand their position in relation to each other and to zero.
Opposites of Numbers
Understanding opposites of numbers is key to mastering concepts on the number line. The opposite of a number is like its mirror image over zero. For each positive number, there is a negative counterpart and vice versa.
  • For example, the opposite of \(-3\) is \(+3\).
  • The opposite of \(-1\) is \(+1\).
  • The opposite of \(+5\) is \(-5\).
Essentially, you can think of opposites as swapping directions across zero while maintaining the same distance from it. This concept helps when plotting multiple points and considering their relations on the real number line.
Number Line Plotting
Number line plotting involves placing numbers on a horizontal line, spaced evenly to show their relative positions. The number line extends infinitely in both the positive and negative directions.
To plot on this line:
  • Start with identifying your smallest number, which might be negative.
  • Place this on the far left and continue placing numbers incrementally to the right until all numbers, including their opposites, are plotted.
In our example, plot starting at \(-5\), then \(-3\), \(-1\), zero, \(+1\), \(+3\), and finally \(+5\). Each point should be equally spaced to accurately reflect their value. This arrangement offers a clear visual understanding of relationships among numbers, making math concepts easier to comprehend.

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