Problem 50 - Algebra 1

Question

Graph the numbers on a number line. Then write two inequalities that compare the two numbers. $$-2.4 \text { and } 3.2$$

Step-by-Step Solution

Answer

The numbers \( -2.4 \) and \( 3.2 \) on a number line visually represent that -2.4 is less than 3.2. Therefore, the two inequalities comparing these numbers are \( -2.4 < 3.2 \) and \( 3.2 > -2.4 \)

Step 1: Represent the Numbers on a Number line

Draw a number line with markings from -3 to 4, considering that the numbers to be plotted are -2.4 and 3.2. Represent -2.4 and 3.2 on the number line. Clearly, -2.4 will be to the left of 3.2.

Step 2: Writing the Inequalities

-2.4 is less than 3.2. Therefore, one possible inequality is \( -2.4 < 3.2 \). The other inequality happens to be the other way round; having 3.2 be greater than -2.4. Hence \( 3.2 > -2.4 \) is the other inequality.

Key Concepts

Understanding Inequalities

Inequalities are mathematical expressions that show the relationship between two values. They help us know whether one number is greater, less, or equal to another.
The common symbols used are:

• \(<\) - less than
• \(>\) - greater than
• \(\leq\) - less than or equal to
• \(\geq\) - greater than or equal to

When you have two numbers, you can form inequalities to express how they compare. For example, given numbers -2.4 and 3.2, we can say -2.4 is less than 3.2, which is written as \(-2.4 < 3.2\). This essentially means -2.4 is located to the left of 3.2 on a number line. Similarly, it’s true to state that 3.2 is greater than -2.4, represented by \(3.2 > -2.4\).
These inequalities clarify the position and magnitude difference between these numbers in an easy-to-understand way.

Graphing Numbers on a Number Line

Graphing numbers on a number line provides a visual representation of their value. This tool makes it easier to see the relationship between numbers, particularly when comparing them.
To graph -2.4 and 3.2, draw a number line that includes these points. Since the value of -2.4 is negative, you'll plot it on the left side of zero. For 3.2, which is positive, plot it to the right side of zero. This layout shows how numbers increase from left to right.
It's important to mark these accurately:

• Locate -2.4 on the left, between -2 and -3.
• Place 3.2 on the right, between 3 and 4.

This visualization aligns with earlier mentioned inequalities and helps solidify the concept of ordering numbers.

Negative and Positive Numbers

Understanding negative and positive numbers is fundamental in math. They describe parts of the number system and how values are assigned and related to zero.

• **Negative numbers** represent values less than zero. They can be associated with losses or depths below a neutral starting point.
• **Positive numbers** indicate values greater than zero. They may symbolize gains or heights above a base.

On a number line, negative numbers extend left from zero, while positive numbers stretch to the right. Whether comparing -2.4 and 3.2 or other numbers, this positioning lets us comprehend how they interact.
Zero acts as a midpoint, enabling clarity between what is negative and positive. In our earlier example, -2.4 is negative, representing a lesser value than the positive 3.2. These distinctions are crucial in recognizing and applying mathematical principles effectively.