Problem 86 - Algebra 1

Question

Graph the numbers on a number line. Then write two inequalities that compare the two numbers. $$-2.8 \text { and } 0.5$$

Step-by-Step Solution

Answer

The numbers graphed on a number line would show '-2.8' to the left of '0.5'. The two inequalities comparing these numbers are \(-2.8 < 0.5\) and \(0.5 > -2.8\).

Step 1: Graph the Numbers on a Number Line

A number line is a visualization tool used to represent numbers and their order. Each point on the line corresponds to a number. Place '-2.8' to the left and '0.5' on the right because '-2.8' is less than '0.5'.

Step 2: Compose the Inequalities

An inequality declares that one quantity is larger or smaller than another. There are two clear inequalities for these two numbers: \(-2.8 < 0.5\) since '-2.8' is less than '0.5'. Also, \(0.5 > -2.8\) as '0.5' is larger than '-2.8'.

Key Concepts

Understanding the Number Line

A number line is an essential mathematical tool that helps in visualizing the relationship between numbers. Imagine a straight line where each point corresponds to a real number. This line is typically oriented horizontally, with smaller numbers on the left and larger numbers on the right.

To graph numbers like \(-2.8\) and \(0.5\) on a number line, consider their positions based on their value. Since \(-2.8\) is a negative number, it will appear to the left of \(0\) on the number line. Meanwhile, \(0.5\) is positive, placing it to the right of \(0\).

• Negative numbers appear to the left of zero.
• Positive numbers appear to the right of zero.

Visualizing numbers in this way makes it easier to compare them, as you can clearly see which values lie further along the line.

Comparing Numbers with Inequalities

Comparing numbers means determining which of the two is greater or lesser. This can be effectively done using inequalities. An inequality shows a relationship where one quantity is larger or smaller than another.

For instance, in the exercise, we have two numbers: \(-2.8\) and \(0.5\). Plotting them on a number line helps us see that \(-2.8\) is less than \(0.5\). Inequalities can represent this comparison mathematically:

• \(-2.8 < 0.5\)
• \(0.5 > -2.8\)

These inequalities mean the same thing but expressed differently. Inequalities are powerful because they make it easy to compare any two numbers and convey the nature of their relationship clearly. They are a fundamental part of understanding mathematical concepts and relationships in everyday situations.

Understanding Negative Numbers

Negative numbers represent values less than zero and are found to the left on a number line. Their placement signifies that they are the opposite of positive numbers. This concept might initially seem abstract, but it is quite straightforward once you grasp the basic rules.

Negative numbers have several key characteristics:

• They increase in absolute size the further they are from zero on the left.
• Adding a negative number means moving left on the number line.
• Subtracting a negative number involves moving right, which is equivalent to adding the positive version of that number.
• Multiplying or dividing two negative numbers results in a positive number.

Understanding these properties makes it easier to use negative numbers in arithmetic operations and compare their values with other numbers. For example, in the exercise, the number \(-2.8\) is a negative number and is smaller than \(0.5\), which is why it appears to the left of \(0.5\) on the number line.