Problem 84 - Algebra 1

Question

Graph the numbers on a number line. Then write two inequalities that compare the two numbers. $$-\frac{1}{2} \text { and } \frac{1}{3}$$

Step-by-Step Solution

Answer

Numbers plotted on a number line: -1/2 positioned on the left side and 1/3 on the right. Two inequalities: \( -\frac{1}{2} < \frac{1}{3} \) and \( \frac{1}{3} > -\frac{1}{2} \).

Step 1: Plotting numbers

Concentration is required to plot two numbers, -1/2 and 1/3 on a number line. The given numbers are fractions and one of these is a negative number. Therefore, it is crucial to position these accurately. A number line usually starts from 0 in the middle, where numbers to the right are positive and to the left are negative. In this scenario, -1/2 will be plotted on the left side of the zero while 1/3 will be positioned on the right.

Step 2: Writing the inequalities

The purpose of the inequalities is to compare two numbers. So, this comparison can be \( -\frac{1}{2} < \frac{1}{3} \) which reads as negative half is less than one third. On the flip side, this can also be written as \( \frac{1}{3} > -\frac{1}{2} \), implying one third is greater than negative half. This is an important consideration as the position of each number on the number line relies on its value.

Key Concepts

Understanding Fractions

Fractions represent parts of a whole. How they are used depends on the numbers at the top and bottom of a fraction, known as the numerator and denominator, respectively. When we see a fraction like \( \frac{1}{3} \), the number 1 is the numerator and symbolizes how many parts are being considered, while 3 is the denominator indicating the number of equal parts the whole is divided into.
A number line helps visualize these fractions by showing where they lie in relation to other numbers. It's important to understand that fractions closer to zero are smaller, while those further from zero are larger. Fractions like \( \frac{1}{3} \) are placed on the right side of zero, indicating they are positive. Differently-sized fractions can be compared by looking at their positions on this number line.

Demystifying Inequalities

Inequalities are expressions that show the relationship between two numbers where they are not equal. The symbols \(<\) and \(>\) are commonly used to show this relationship. For example, in \( -\frac{1}{2} < \frac{1}{3} \), the symbol \(<\) signifies that \(-\frac{1}{2}\) is less than \(\frac{1}{3}\). This means if you plot them on a number line, \(-\frac{1}{2}\) will be found to the left of \(\frac{1}{3}\). Similarly, \(\frac{1}{3} > -\frac{1}{2}\) illustrates the reverse relationship.

• Inequalities help us compare numbers, including whole numbers, fractions, and even negative numbers.
• They provide a simple way to express that one number is greater than or less than another.

Understanding inequalities is crucial for making quick and accurate decisions when comparing various numerical values.

Navigating Negative Numbers

Negative numbers can be confusing at first, but they follow the same logic as positive numbers with their own unique rules. They are positioned left of zero on a number line. The further left they are, the smaller they are considered. For instance, -1 is less than 0 because it is further left.
Negative numbers such as \(-\frac{1}{2}\) indicate something less than zero. This concept is essential for understanding real-world scenarios involving loss, debt, or temperatures below freezing.

• Negative numbers are used in everyday situations like banking (overdrafts) or measuring temperatures (below zero).
• They can be tricky when combined with fractions, but the same rules of inequalities apply.

By becoming comfortable with these concepts, you'll be able to work effectively with both positive and negative numbers.