Problem 65 - Beginning and Intermediate Algebra

Question

Select the lesser of the two given numbers. \(-\frac{2}{3},-\frac{1}{4}\)

Step-by-Step Solution

Answer

-\frac{2}{3} is the lesser number.

Step 1: Understand the Problem

We need to find which of the two numbers \(-\frac{2}{3}\) and \(-\frac{1}{4}\) is lesser. In other words, we are looking for the number that is further to the left on the number line.

Step 2: Compare Negative Fractions

A negative fraction that has a larger absolute value is considered to be lesser because it is further to the left on the number line.

Step 3: Determine Absolute Values

Find the absolute values of the given numbers. Absolute value of \(-\frac{2}{3}\) is \frac{2}{3}\ and absolute value of \(-\frac{1}{4}\) is \frac{1}{4}\.

Step 4: Compare Absolute Values

Since \frac{2}{3} > \frac{1}{4}, \(-\frac{2}{3}\) will be lesser than \(-\frac{1}{4}\).

Step 5: Conclusion

From the comparison, \(-\frac{2}{3}\) is the lesser number.

Key Concepts

Absolute Value

When working with negative fractions, understanding the concept of absolute value is crucial. The absolute value of a number is its distance from zero on the number line, irrespective of its direction. For example, the absolute value of both \(-2\/3\) and \(+2\/3\) is \(+2\/3\). Similarly, the absolute value of \(-1\/4\) and \(+1\/4\) is \(+1\/4\). Absolute value helps us compare magnitude without considering the sign. This is important because a larger absolute value means a number is further from zero.

Number Line

Visualizing fractions on the number line is a powerful tool for comparing them. On a number line, numbers increase as you move to the right and decrease as you move to the left. Negative numbers are on the left of zero, and positive numbers are on the right. When you place \(-2\/3\) and \(-1\/4\) on a number line, \(-2\/3\) appears further to the left compared to \(-1\/4\). This indicates that \(-2\/3\) is less than \(-1\/4\). By using a number line, you can directly see that a negative fraction with a larger absolute value is 'lesser' because it's further to the left.

Fraction Comparison

Comparing fractions involves looking at both numerators and denominators. When dealing with negative fractions, comparison often involves absolute values. First, find the absolute values of the fractions. For \(-2\/3\), the absolute value is \(+2\/3\), and for \(-1\/4\), it's \(+1\/4\). Since \(+2\/3\) is greater than \(+1\/4\), it follows that \(-2\/3\) should be less than \(-1\/4\) because it is further left on the number line. Therefore, understanding both absolute values and their positions on the number line simplifies the process of comparing negative fractions.