Problem 28 - Algebra 1: Concepts and Skills

Question

Graph the numbers on a number line. Then write two inequalities that compare the numbers. \(8,-8\)

Step-by-Step Solution

Answer

The two inequalities comparing the numbers 8 and -8 are \(8 > -8\) and \(-8 < 8\).

Step 1: Draw a number line

Firstly, draw a straight horizontal line, then indicate the point 0 (zero) anywhere on the line. On the left side of zero, marks should be placed at equal intervals, which represent negative numbers. Similarly, marks should be placed on the right side of zero, representing positive numbers.

Step 2: Place the given numbers on the number line

Locate and mark the given numbers on the number line. Here it gives us two numbers, 8 and -8. So find where 8 would be on the positive side (right side of zero) and where -8 would be on the negative side (left side of zero).

Step 3: Sketch arrows pointing to the direction of the numbers.

Draw arrows that point towards the positions of 8 and -8; this shows that these are the numbers being considered.

Step 4: Write two inequalities

Comparing the two numbers, we can say that 8 is greater than -8 which can be written as \(8 > -8\) or alternately -8 is less than 8 which can be written as \(-8 < 8\).

Key Concepts

Number Line

A number line is a great tool to visually represent numbers. Imagine it as a straight horizontal line. Most importantly, it has a special point called zero, located at the center. The number line continues infinitely in both directions. Each point on this line corresponds to a unique real number.

• On the right side of zero, the numbers are positive.
• On the left side of zero, the numbers are negative.

To draw a number line manually, first find a place to indicate zero. Once you've marked zero, place equal marks at consistent intervals on both sides to show other numbers. This consistency will help you accurately assess the position of specific numbers, such as in exercises where you may need to graph numbers like 8 and -8. The number line helps compare numbers by observing their positions relative to zero and each other.

Negative Numbers

Negative numbers are found to the left of zero on the number line. They represent values less than zero and indicate a deficit or a decrease in quantity. For example, -1, -2, -3, and so on. These numbers can be thought of as moving backwards from zero.
When working with negative numbers on a number line, it helps to remember:

• The further left a number is, the smaller its value. Therefore, -8 is smaller than -3.
• Negative numbers are always less than positive numbers. For instance, -8 is less than 8, which can symbolically be written as deficit\(-8 < 8\).

Visualizing negative numbers using the number line makes it easier to understand these concepts, especially when comparing them with other numbers.

Positive Numbers

Positive numbers are located to the right of zero on the number line. They are the standard numbers that we use in everyday counting: 1, 2, 3, and so on. These numbers indicate a gain or an increase in quantity compared to zero.
Important aspects to consider about positive numbers include:

• The further right a number is, the larger its value. Thus, 8 is greater than 3.
• No matter how small, a positive number is always greater than any negative number. For example, 8 is greater than -8 or \(8 > -8\).

Using the number line, we can easily place and recognize positive numbers, which is especially helpful for visual learning and graph-related exercises.