Problem 19 - Algebra 1

Question

Graph the numbers on a number line. Then write two inequalities that compare the two numbers. $$-6.4 \text { and }-6.3$$

Step-by-Step Solution

Answer

The two numbers, -6.4 and -6.3, are represented on the number line with -6.4 to the left of -6.3. In terms of inequalities, -6.4 < -6.3 and -6.3 > -6.4.

Step 1: Draw the Number Line

Start by drawing a horizontal line, which represents the number line. Then, place tick marks on the line to represent numbers. Make sure the tick marks for -6.4 and -6.3 are appropriately positioned. Typically, numbers to the left on a number line are smaller than those to the right. So, -6.4 should be to the left of -6.3 on the number line.

Step 2: Mark the Numbers

Next, mark the positions of -6.4 and -6.3 on the number line. You can label these points for clarity. This visual representation will aid in understanding their relation.

Step 3: Write the Inequalities

Now, use the symbols '<' (less than) or '>' (greater than) to create two inequalities. Recall that -6.4 is less than -6.3, and -6.3 is greater than -6.4. Therefore, the inequalities are -6.4 < -6.3 and -6.3 > -6.4.

Key Concepts

Number Line

A number line is a simple and powerful way to represent numbers in a straight line, where each point corresponds to a number. It extends infinitely in both directions and usually features a series of tick marks indicating integer values.

• To graph numbers like \(-6.4\) and \(-6.3\), we start by determining where each lies on the number line.
• Numbers decrease as you move from right to left.
• Thus, \(-6.4\) is placed to the left of \(-6.3\).

A number line helps us see the difference and relationship between numbers quickly. It's an essential tool for visualizing mathematical concepts, especially when dealing with both positive and negative values.

Comparing Numbers

When comparing numbers, especially negative ones, it's vital to understand the role of their position on the number line.

• If a number is located to the left of another number, it is considered smaller.
• Conversely, if it is to the right, it is considered larger.

For example, with the numbers \(-6.4\) and \(-6.3\), the number \(-6.4\) is positioned to the left indicating it is smaller. Thus, we write the inequality \(-6.4 < -6.3\). The opposite holds where \(-6.3\) is greater, so \(-6.3 > -6.4\). These inequalities describe their relative positions on the number line, translating visual distance into mathematical relationships.

Negative Numbers

Negative numbers are those less than zero and are typically found on the left side of the number line.

• As numbers become more negative, their value decreases. Thus, \(-6.4\) is less than \(-6.3\) because it is further left on the number line.
• It's important to remember that a negative number becomes 'larger' as it moves closer to zero.

The concept might seem counterintuitive at first, but once we see how negative numbers are placed and compared on the number line, it becomes easier to understand their behavior and use them in inequalities. Always pay attention to the direction on the number line, as it determines the sign of the inequality.