Problem 71 - Algebra A Combined Function

Question

Insert \(<,>,\) or \(=\) in the appropriate space to make each statement true. $$ \left|-\frac{5}{8}\right| \quad\left|\frac{5}{8}\right| $$

Step-by-Step Solution

Answer

The correct symbol is \( = \).

Step 1: Understand Absolute Values

The absolute value of a number is its distance from zero on the number line, without considering direction. Therefore, the absolute value of any positive or negative number is always positive. For any real number \( x \), the absolute value is denoted as \( |x| \).

Step 2: Calculate Absolute Values

Calculate the absolute values of the given numbers. First, we find \( \left| -\frac{5}{8} \right| = \frac{5}{8} \) since the absolute value of a negative number is the number itself without the negative sign. Similarly, for the second number, \( \left| \frac{5}{8} \right| = \frac{5}{8} \).

Step 3: Compare the Absolute Values

Since both absolute values calculated in Step 2 are \( \frac{5}{8} \), they are equal to each other. Therefore, we place an equals sign \( = \) between them to reflect this equivalence.

Key Concepts

comparing fractions

Comparing fractions is a fundamental skill that helps you understand the magnitude of different values. To compare fractions:

• Ensure fractions have the same denominator. This makes it easy to directly compare numerators.
• If denominators are different, find a common denominator. Adjust the numerators accordingly, and then compare.
• Another method is converting fractions to decimal form for easier comparison.

In our exercise, we compared \(-\frac{5}{8}\) and \(\frac{5}{8}\) using absolute values, yielding \(\frac{5}{8}\) for both, showing that they are equivalent. Understanding this principle helps in various mathematical contexts, not just when dealing with absolute values.

real numbers

Real numbers encompass all the values on the number line, including integers, fractions, and irrational numbers. Here’s what you need to know:

• They can be positive, negative, or zero.
• They include fractions and decimals, like \(\frac{5}{8}\).
• Each real number has an absolute value which is its distance from zero on the number line.

In the exercise, both numbers given in fraction forms, \(-\frac{5}{8}\) and \(\frac{5}{8}\), are real numbers. Converting to absolute values simplifies comparisons and calculations on the number line.

number line

A number line is a visual representation of numbers in a straight line where each point corresponds to a number. Here’s how it helps:

• The center, labeled as zero, divides positive and negative numbers.
• It helps to easily visualize distances or differences between numbers.
• Absolute values represent distances from zero, ignoring direction.

In solving the original exercise, the number line aids in understanding that \(-\frac{5}{8}\) and \(\frac{5}{8}\) are equal in absolute terms, as they both lie \(\frac{5}{8}\) units away from zero.