Problem 10 - Introductory and Intermediate Algebra for College Students 4th

Question

Start by drawing a number line that shows integers from \(-5\) to \(5 .\) Then graph each real number on your number line. $$5$$

Step-by-Step Solution

Answer

The number line extends from -5 to 5, and the real number 5 is clearly marked on the number line.

Step 1: Drawing the number line

The first task is to draw a horizontal straight line. This will serve as your number line.

Step 2: Marking the integers

Starting from a point in the middle of the line, draw short, equally spaced vertical lines above the horizontal line. Start marking these lines with integers, starting from 0 at the middle and then going over to both the left and right up to -5 and 5 respectively.

Step 3: Plot the Number

Now, draw a dark dot on the vertical line marked as 5. This denotes the real number 5 on your number line.

Key Concepts

Graphing Real Numbers

Understanding how to graph real numbers is fundamental in visualizing and grasping the concept of numerical value and its position in relation to other numbers. Imagine a straight, horizontal line. This is not just a line but a scale representing real numbers, with the center typically marked as zero. The line extends infinitely in both directions, serving as a visual representation of all real numbers.

To graph a real number, such as the number 5, you simply need to locate its position on this infinite scale. Think of the number line as the home address for numbers. Just like finding a house in a street, the number line allows you to locate the 'home' of any real number. The key is to remember that each point on the line corresponds to a unique real number, which means the number 5 has a distinct spot where it resides. By placing a point above the appropriate mark on the number line, you effectively graph the real number.

Plotting on Number Line

Plotting numbers on a number line is a visual exercise that helps you understand the concept of ordering and magnitude. When you're plotting on a number line, you are essentially organizing numbers according to their size and sequence.

For integers, which are whole numbers without any fractions or decimals, this is a straightforward process. Starting from zero, which is our neutral point, the integers on the right are positive, increasing in value as they move away from zero. On the left, they are negative, decreasing in value. To plot any integer, find its numerical value on this spectrum and place a mark or dot directly above the line. For instance, when plotting the number 5, you would count five spaces to the right of zero and put your dot there. This method promotes a clear and tangible way to compare the size of integers and can be particularly helpful for visual learners.

Integers in Algebra

Integers play a crucial role in algebra as they are the building blocks for more complex mathematical concepts. Algebra often involves finding unknown values, and integers help us to understand and solve these problems because they follow consistent and predictable patterns.

An understanding of how to work with integers on a number line lays the groundwork for tackling algebraic expressions and equations. For example, if you're given a simple equation like \(x + 3 = 8\), you can visualize this on a number line by starting at the number 3 and moving to the right (since you are adding) until you reach 8. The number of steps you take will give you the value of the integer x, which in this case is 5. By mastering the basics of integer manipulation on number lines, you prepare yourself to handle more sophisticated problems in algebra.