Problem 9 - Introductory Algebra for College Students

Question

Start by drawing a number line that shows integers from \(-5\) to \(5 .\) Then graph each real number on your number line. $$2$$

Step-by-Step Solution

Answer

The number 2 has been plotted two divisions to the right from the 0 on the number line.

Step 1: Draw Number Line

Draw a straight horizontal line. This represents the number line. Mark it starting from -5 on the left end and ending with 5 on the right end. Make nine equal partitions between these two numbers to represent the individual numbers from -4 to 4.

Step 2: Mark Zero

Mark the central division on the number line which represents 0. This separates the positive integers from the negative integers.

Step 3: Plot number

The number given is 2. Since it's positive, plot a dot on the second division on the right of 0 on the number line.

Key Concepts

Integers

Understanding integers is key when working with number lines. Integers include whole numbers and their negative counterparts. This means all the numbers without fractions or decimals, such as

• negative numbers like \(-1, -2, -3\),
• zero \(0\),
• and positive numbers like \(1, 2, 3\).

Integers are often represented on a number line, which helps us visualize their order and spacing.
Negative integers are shown to the left of zero, while positive integers are to the right. Zero serves as the central point that divides these two sets. This visualization makes it easier to perform operations like addition and subtraction or understand ordering and magnitude among whole numbers.

Real Numbers

Real numbers are all the numbers you can find on the number line. This includes every integer, but also all the numbers that can be written with a decimal point, such as fractions and irrational numbers.
Examples of real numbers include:

• positive numbers like \(2.5\)
• negative numbers like \(-3.7\)
• irrational numbers like \(\pi\)
• and fractions like \(\frac{1}{2}\).

Real numbers fill the gaps between integers on the number line. They allow us to accurately express values more precisely, which is particularly useful in various real-world applications, like measuring length or expressing probabilities. By plotting real numbers on the number line, you get a clear visual understanding of their relationship to each other.

Graphing Numbers

Graphing numbers on a number line is a practical way to visualize their relationships. When you graph numbers, you're placing them as points on a line. Each point represents a unique value, making it simple to compare their relative sizes. The process often goes like this:

• First, draw a horizontal line, marking equal intervals along it.
• Next, label specific points that correspond to integers or any real numbers.
• Finally, place a dot on the line at the position that matches each number you're graphing.

For example, to graph the number \(2\) on a number line that shows integers from \(-5\) to \(5\), plot it on the second position to the right of zero. This method is straightforward and ensures you can visualize both the order and the spacing of numbers. Using a number line can also help when solving equations or inequalities, as it offers a clear view of how numbers compare to each other.