Problem 24 - Algebra 1

Question

Graph the numbers on a number line. Then write two inequalities that compare the two numbers. \(-2.7\) and \(\frac{3}{4}\)

Step-by-Step Solution

Answer

-2.7 < 3/4 and 3/4 > -2.7

Step 1: Representation of numbers on the number line

To graph the numbers \(-2.7\) and \(\frac{3}{4}\) on a number line, start from zero. Move 2.7 units to the left for \(-2.7\) and a little less than 1 unit to the right for \(\frac{3}{4}\). Make sure to label each number below the point on the line.

Step 2: Comparing numbers by inspection

From the number line, it can be observed that \(-2.7\) is to the left of zero and \(\frac{3}{4}\) is to the right. Encompassing that \(a < b\) for numbers \(a\) and \(b\) on a number line if \(a\) is to the left of \(b\), it can be deduced that \(-2.7 < \frac{3}{4}\). Reversing the inequality gives \(\frac{3}{4} > -2.7\)

Key Concepts

Number Line

A number line is a straight line on which numbers are placed at equal intervals. It's a useful tool for visualizing the relationship between numbers, especially when dealing with inequalities. The number line includes positive numbers to the right of zero, negative numbers to the left, and zero itself.

• Positioning Numbers: Each point on the number line corresponds to a real number. Negative numbers are positioned left of zero, while positive numbers are right of zero.
• Intervals: The line is usually marked with intervals, sometimes smaller marks between them for precision.

In practice, to place a number like \(-2.7\) on the number line, you start at zero and move 2.7 units to the left. For \(\frac{3}{4}\), you start at zero again but move a little less than a full unit to the right. This helps visualize which number is greater.

Comparing Numbers

Comparing numbers is the process of determining which is larger or smaller. When numbers are placed on a number line, their positions can help us see their comparative sizes.

• Left and Right Rule: On a number line, a number located further to the right is greater than one further to the left. In our example, \(-2.7\) is on the left of \(\frac{3}{4}\), indicating \(-2.7 < \frac{3}{4}\).
• Understanding Inequalities: Inequalities are mathematical statements that show how numbers relate to each other in size. Two common inequality symbols are \( < \) and \( > \).

Thus, using a number line, inequalities can be easily visualized and written as \(a < b\) or \(b > a\). This process allows us to appreciate the relative values of \(-2.7\) and \(\frac{3}{4}\).

Graphing Numbers

Graphing numbers on a number line is a method of placing specific values on a line to show their relation visually. This is helpful in understanding where numbers lie in the context of each other.

• Marking Points: Numbers, such as \(-2.7\) and \(\frac{3}{4}\), are graphed by placing a dot at the appropriate point on the line.
• Labeling: Ensure each point is clearly labeled to avoid confusion. This is particularly important when the numbers are close together or when precision is needed.

Graphing helps in quickly identifying inequalities. By seeing the position of \(-2.7\) and \(\frac{3}{4}\) on the number line, we can immediately conclude their comparative sizes and write the inequalities like \(-2.7 < \frac{3}{4}\). Graphing is not just about positioning—it tells a story of number relationships.