Problem 79 - Elementary and Intermediate Algebra

Question

Insert one of the symbols \(>,<,\) or \(=\) in the blank to make each statement true. $$ 1 \quad \text{_____}\quad\left|-\frac{15}{16}\right| $$

Step-by-Step Solution

Answer

1 > \left| -\frac{15}{16} \right|.

Step 1: Evaluate the Absolute Value

To solve the inequality, first evaluate the absolute value expression. Recall the property that the absolute value of a number is its distance from zero on the number line, regardless of its sign.The given expression is \( \left| -\frac{15}{16} \right| \). The absolute value removes the negative sign, making this value \( \frac{15}{16} \).

Step 2: Compare the Two Numbers

Now, compare the two values: 1 and \( \frac{15}{16} \).Convert both numbers into a format that is easy to compare:- 1 is the same as \( \frac{16}{16} \).- From Step 1, we have \( \frac{15}{16} \).So, compare \( \frac{16}{16} \) with \( \frac{15}{16} \).

Step 3: Determine the Relationship

To find out which is larger, compare the two fractions:Since \( \frac{16}{16} \gt \frac{15}{16} \), it follows that 1 is greater than \( \frac{15}{16} \).

Step 4: Insert the Correct Symbol

Insert the correct symbol based on the comparison from Step 3. We need to insert the greater than sign (\( > \)) into the blank space to indicate that 1 is greater than \( \left| -\frac{15}{16} \right| \).

Key Concepts

Comparison of Numbers

When comparing numbers, it is important to think about where each number is located on the number line. A number further to the right on the number line is always larger than a number further to the left. This idea is straightforward when dealing with whole numbers but can require some extra steps with fractions or absolute values.
Evaluating absolute values can change a negative number into a positive one. For example, the absolute value of \(-\frac{15}{16}\) is \(\frac{15}{16}\) since absolute value measures distance from zero. Once absolute values are accounted for, you can more easily compare numbers by ensuring they are in the same form.

Inequalities

Inequalities help us understand the relationship between two values. By using a few key symbols, we can show if a number is greater than, less than, or equal to another number.

• "\(>\)" stands for "greater than"
• "\(<\)" stands for "less than"
• "\(=\)" means "equal to"

In this exercise, we needed to determine whether 1 is greater than, less than, or equal to \(\left|-\frac{15}{16}\right|\). By evaluating the absolute value and comparing it as a positive fraction, we are able to insert the correct inequality symbol. Understanding inequalities is crucial because it helps you solve problems that require comparison or determining limits.

Fraction Comparison

Comparing fractions is a necessary skill, especially when dealing with absolute values and inequalities. Fractions represent parts of a whole, and to compare them, it's often useful to express them with a common denominator.
Here, we have 1 and \(\frac{15}{16}\). Converting 1 to a fraction with the same denominator as \(\frac{15}{16}\) allows easy comparison. By expressing 1 as \(\frac{16}{16}\), we can directly see that \(\frac{16}{16}\) is greater than \(\frac{15}{16}\).
It’s a simple but powerful technique to convert numbers into like terms—which is often converting whole numbers to fractions or finding equivalent fractions with common denominators. This ensures an accurate comparison, making it easy to determine which fraction is larger or smaller.