Problem 11 - 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. $$-5$$

Step-by-Step Solution

Answer

The number \(-5\) has been graphed on the number line ranging from \(-5\) to \(5\).

Step 1: Draw the number line

Create a straight line. This will be the basis for the number line. On this line, equally space marks will represent integers. Be sure to clearly label each mark with its corresponding integer number, ranging from \(-5\) to \(5\).

Step 2: Graph the number

For this case, the real number to graph is \(-5\). Find \(-5\) on the number line and indicate it. A common way is to draw a dot or a small circle earlier placed on the number \(-5\).

Key Concepts

Integers

Integers are the numbers we use for counting and ordering in everyday life. They include positive numbers, negative numbers, and zero. Integers can be visualized easily on a number line, where each point is an integer and evenly spaced.

• Positive integers: Numbers greater than zero, like 1, 2, and 3.
• Negative integers: Numbers less than zero, like -1, -2, and -3.
• Zero: Neither positive nor negative, it sits in the middle of the number line.

Understanding integers helps us grasp the concept of whole numbers and their positions relative to each other. On a number line from -5 to 5, you'll see numbers increasing from left to right.

Real Numbers

Real numbers are an essential part of mathematics, encompassing all the numbers you can think of, including both rational and irrational numbers.

• Rational numbers: Numbers that can be expressed as a fraction, such as \( \frac{1}{2} \) or -3.
• Irrational numbers: Numbers that cannot be accurately expressed as a simple fraction, like \( \pi \) or \( \sqrt{2} \).

Real numbers appear continuously on the number line and fill in all the gaps between integers. By understanding real numbers, you can better navigate problems beyond simple counting, such as measuring distances or solving equations.

Graphing Numbers

Graphing numbers on a number line is an effective way to visualize their relationships. The number line is a straight line with markings that represent numbers.

• Identifying integers: Clearly marked and evenly spaced at intervals along the line.
• Placing real numbers: Real numbers, such as decimals or fractions, can be plotted between these integer points.
• Using symbols: Often dots or small circles are used to indicate specific numbers.

When graphing, start with a clear and labeled line, marking each integer. Then, plot the given real number, such as -5, by placing a dot right on the corresponding mark. This visual representation helps in understanding number placement and comparisons.