Problem 17 - College Algebra and Calculus: An Applied Approach

Question

Plot the two real numbers on the real number line and place the appropriate inequality symbol \((<\) or \(>)\) between them.\(\frac{5}{6}, \frac{2}{3}\)

Step-by-Step Solution

Answer

\(\frac{5}{6} > \frac{2}{3}\)

Step 1: Understand the fractions

We are given two fractions \(\frac{5}{6}\) and \(\frac{2}{3}\). Both these fractions are less than 1 but more than 0. We can also convert \(\frac{2}{3}\) into a fraction with denominator 6 to compare them more easily. That would be \(\frac{4}{6}\). Now, it's clearer that \(\frac{5}{6}\) is greater than \(\frac{2}{3}\) in value.

Step 2: Plot on the number line

Draw a number line and mark 0 and 1 clearly. Since both the fractions are between 0 and 1, divide the section between 0 and 1 into 6 equal parts because the denominators of given fractions are 3 and 6. The fraction \(\frac{5}{6}\) will be located at the fifth division from zero, and \(\frac{2}{3}\) or \(\frac{4}{6}\) will be located at the fourth division from zero.

Step 3: Place the inequality symbol

Now that we have the numbers placed on the number line, we can easily see that \(\frac{5}{6}\) is greater than \(\frac{2}{3}\). Therefore, we place the '>' symbol between them. So \(\frac{5}{6} > \frac{2}{3}\)

Key Concepts

Comparing Fractions

Fractions are numerical quantities that represent a part of a whole. To compare fractions, one of the simplest methods is to convert them to a common denominator. When fractions share the same denominator, the comparison becomes straightforward, as you can directly compare the numerators. The fraction with the larger numerator is the larger fraction. For example, to compare \(\frac{5}{6}\) and \(\frac{2}{3}\), we convert \(\frac{2}{3}\) to an equivalent fraction with 6 in the denominator, obtaining \(\frac{4}{6}\). Comparing the numerators, 5 and 4, it's clear that \(\frac{5}{6}\) is greater than \(\frac{4}{6}\), or in other words, \(\frac{5}{6}\) is greater than \(\frac{2}{3}\).

Another method of comparison is by cross-multiplication. Multiply the numerator of the first fraction by the denominator of the second, and do the same for the numerator of the second fraction and the denominator of the first. This method is particularly useful when the denominators are not easily comparable or when working with larger numbers. For our example, this isn’t necessary, but it’s a handy tool to have for more complex comparisons.

Inequality Symbols

Inequality symbols are mathematical notations used to represent the relationship between two values. The symbol \(>\) means 'greater than', while \(<\) means 'less than'. When comparing two numbers, the wider part of the symbol always points to the larger value. For instance, when we say \(5 > 3\), we mean that 5 is greater than 3. Conversely, \(3 < 5\) communicates that 3 is less than 5.

Using inequality symbols helps clearly define the relationship between fractions, as we’ve seen with \(\frac{5}{6}\) and \(\frac{2}{3}\). After comparing the fractions and determining that \(\frac{5}{6}\) is larger, we use the \(>\) symbol to express this: \(\frac{5}{6} > \frac{2}{3}\). Inequality symbols are essential tools in mathematics that help us describe and solve equations, as well as understand the relationships between different quantities.

Number Line Representation

A number line is a visual representation of numbers on a straight line, where each point corresponds to a number. In the context of fractions, a number line can help illustrate the relative sizes of different fractions. To plot fractions on a number line, first identify the range that will encompass the fractions—in this case, between 0 and 1. Then, divide the line into equal segments based on the denominators of the fractions. For \(\frac{5}{6}\) and \(\frac{2}{3}\), we divide the section between 0 and 1 into 6 equal parts because 6 is a common denominator for both fractions.

The fraction \(\frac{5}{6}\) will be placed at the fifth marker from 0, while \(\frac{2}{3}\), or its equivalent \(\frac{4}{6}\), will be placed at the fourth marker. This visual approach immediately shows that \(\frac{5}{6}\) is further to the right than \(\frac{2}{3}\), confirming that the former is larger. Representing fractions on a number line not only assists in comparing them but also improves our understanding of the concept of fractions and their magnitudes in relation to the whole and to each other.