Problem 71 - Elementary and Intermediate Algebra

Question

Insert one of the symbols \(>,<,\) or \(=\) in the blank to make each statement true. $$ |-1.1| \quad \text{_____}\quad1.2 $$

Step-by-Step Solution

Answer

|-1.1| < 1.2

Step 1: Understand the Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. Therefore, it is always non-negative. For example, the absolute value of \(-1.1\) is 1.1, as it disregards the negative sign.

Step 2: Compare the Numbers

Now that we know that \(|-1.1| = 1.1\), we need to compare 1.1 and 1.2. We can see that 1.1 is less than 1.2.

Step 3: Insert the Correct Symbol

Since 1.1 is less than 1.2, we use the symbol \(<\) to indicate this. Therefore, the statement should be \(|-1.1| < 1.2\).

Key Concepts

Number Comparison

Number comparison involves determining which of two numbers is larger, smaller, or if they are equal. In our exercise, we look at the absolute value of \(-1.1\) and compare it with 1.2.

To make accurate comparisons:

• Place the numbers on a number line and visualize their positions.
• Identify which number is farther to the left; this number is smaller.
• If two numbers are identical, they are equal.

Using these steps allows us to say that 1.1 is less than 1.2 because 1.1 is located to the left of 1.2 on a number line. Therefore, in our exercise, we find that \(|-1.1|\) is less than 1.2, so \(1.1 < 1.2\).

Mastering number comparison is crucial for solving math problems accurately and confidently.

Distance on the Number Line

The distance on the number line is vital in understanding absolute values. It helps represent numbers as points on a line extending infinitely in both positive and negative directions.

Absolute value measures how far a number is from zero on this line. It doesn't matter if the number is positive or negative; the distance is always positive. For example:

• The absolute value of \(-1.1\) is 1.1 because it is 1.1 units away from zero.
• Similarly, the absolute value of 1.1 is also 1.1.

Visualizing numbers on a number line helps to compare distances and understand the concept of absolute value more deeply.

Whenever you encounter absolute values, remember that you're dealing with distances, not directions.

Inequality Symbols

Inequality symbols are mathematical tools used to compare quantities. The common symbols are:

• \(>\) - greater than
• \(<\) - less than
• \(=\) - equal to

These symbols simplify the representation of number relationships.

When comparing numbers' sizes, as in our exercise, the correct inequality symbol shows the relationship. Here, since \(|-1.1|\) (which is 1.1) is smaller than 1.2, we use \(<\) to show less than, making the expression \(|-1.1| < 1.2\).

Understanding and correctly using inequality symbols is crucial for solving mathematical problems that involve comparisons and calculations. Always ensure you're using the right symbol to reflect the correct relationship.