Problem 26 - Algebra 1

Question

Graph the numbers on a number line. Then write two inequalities that compare the two numbers. $$-1 \frac{5}{6} \text { and }-1 \frac{7}{9}$$

Step-by-Step Solution

Answer

The numbers plotted on a number line show that -11/6 is less than -16/9 and, conversely, -16/9 is greater than -11/6. Therefore, the inequalities are \(-\frac{11}{6} < -\frac{16}{9}\) and \(-\frac{16}{9} > -\frac{11}{6}\).

Step 1: Convert Mixed Numbers to Improper Fractions

Firstly, it’d be easier comparing the two numbers if they were both in a similar format. Since they are mixed fractions, it’s advisable to convert them into improper fractions. For -1 5/6, to convert, multiply the whole number by the denominator and add the numerator, giving you -11/6. Similarly, for -1 7/9, this gives -16/9.

Step 2: Graph the Numbers on a Number Line

Draw a number line and plot the two numbers. Since these are negative numbers, they will be located to the left of zero. The larger a negative number, the farther left it will be from zero. Depending on conversion, -11/6 will occupy a space more to the left than -16/9.

Step 3: Write Two Inequalities That Compare The Two Numbers

From the number line plot, -11/6 is less than -16/9 and -16/9 is greater than -11/6. Therefore, the inequalities are \(-\frac{11}{6} < -\frac{16}{9}\) and \(-\frac{16}{9} > -\frac{11}{6}\).

Key Concepts

Understanding Number Lines in Inequalities

A number line is a simple yet powerful tool for visualizing numbers, especially when comparing inequalities. When you graph numbers on a number line, you place them in order from smallest on the left to largest on the right. This visual representation helps in understanding which numbers are greater or lesser.

When dealing with negative numbers, it's crucial to remember that a number farther to the left is smaller. For instance, on a number line, -2 is to the left of -1, indicating that -2 is smaller than -1.

In the context of the exercise, plotting values such as i.e., -11/6 and -16/9, on a number line can show their relative positions and help us easily see and write inequalities like i.e., \(-\frac{11}{6} < -\frac{16}{9}\). This method of placement makes it easier to compare fractions and mixed numbers.

Demystifying Improper Fractions

Improper fractions can often be confusing, but they are quite straightforward once you break them down. An improper fraction has a numerator larger than its denominator. This indicates the fraction is greater than 1 (or less than -1, if negative). For example, \(\frac{11}{6}\) is an improper fraction because 11 is greater than 6.

To convert a mixed number into an improper fraction, multiply the whole number by the fraction's denominator and then add the numerator. So, for the number -1 \(\frac{5}{6}\), it becomes \(-\frac{11}{6}\).The process is the same for a negative number but with attention to the sign.

This conversion is useful when comparing two numbers because doing so places them into a similar format, making it clearer which might be greater or lesser, especially on a number line.

Exploring Mixed Numbers

Mixed numbers consist of a whole number and a fractional part. They can sometimes appear less daunting compared to improper fractions, but both represent the same value. For example, -1 \(\frac{5}{6}\) is a mixed number indicating that you have -1 whole, and you take away another \(\frac{5}{6}\).

Mixed numbers are often popular in everyday math because they more clearly represent the quantity without requiring an understanding of larger numerators.

However, for mathematical operations like comparing size, converting to improper fractions is beneficial. It allows for easier calculations and alignment on a number line, helping you write inequalities accurately. Whether you choose improper fractions or mixed numbers, understanding their equivalency is key to mastering fractions and their applications in inequalities.