Problem 11 - 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 real number -5 is plotted on the farthest left of the number line, at the point labeled -5.

Step 1: Draw a Number Line

First, draw a straight horizontal line. This will represent the number line. Mark an arbitrary point on this line as 0. This is your base point. Now mark equal intervals on either side of this 0 point. Going to the right, mark 5 points and label them as 1, 2, 3, 4, 5 respectively. Similarly, going to the left of the 0 point, mark five points and label them as -1, -2, -3, -4, -5 respectively.

Step 2: Plot the Given Number

The given real number is -5. This corresponds with the point on the farthest left of the number line (since the line was drawn from -5 to 5). Therefore, plot or mark the point labeled as -5 on your number line and highlight it.

Step 3: Verify the Placement

Confirm if -5 is correctly placed on the number line. Since all the numbers to the right of -5 are greater and all the numbers to its left (if any) are lesser, the placement of -5 on the number line is correct.

Key Concepts

Understanding Integers

Integers are whole numbers that can be both positive, negative, or zero. This means they include numbers like -3, 0, and 2. Integers are useful because they allow us to easily represent real-world situations, such as debts or elevations above and below sea level.
They can be depicted on a number line, which helps visualize the order and spacing of these numbers.

• Positive integers, such as 1, 2, and 3, are placed to the right of zero on a number line.
• Negative integers, like -1, -2, and -3, are placed to the left of zero.
• Zero itself serves as a central reference point.

Number lines are helpful for graphing and comparing integers, showing how they relate to one another. For instance, -5 is less than -3 because it appears to the left of -3 on the number line.

Exploring Real Numbers

Real numbers are a broad category that includes not only integers but also fractions, decimals, and irrational numbers like \( \sqrt{2} \) and \( \pi \). Real numbers can be plotted on a number line as well, filling in all the spaces between integers.
This means that while integers like -2 or 1 are specific points, the real numbers encompass every possible value in between, like -2.5 or 0.333...

• Every integer is a real number, but not every real number is an integer.
• Real numbers can be rational (such as 1/2 or 0.75) or irrational (such as \( \sqrt{2} \)).

Number lines can accommodate the entire span of real numbers, making them ideal for visually representing data that includes decimals and fractions alongside whole numbers.

Basics of Graphing on a Number Line

Graphing on a number line involves marking a specific point that corresponds to a particular number. This process is straightforward but essential for visually understanding and interpreting numbers.
A number line typically starts with zero at the center, with positive numbers extending to the right and negative numbers to the left.

• To graph an integer, like -5, on a number line, start from zero and move left or right to the correct position.
• For example, to place -5 on a line stretching from -5 to 5, identify the leftmost point as -5 and mark it to highlight this value.

You can visualize the relative size and order of numbers easily by plotting them on a number line. This helps in comparisons, as graphing gives a clear picture of which numbers are larger or smaller, as well as how far apart they are.