Problem 32 - Algebra 2

Question

Find each absolute value. $$ |11| $$

Step-by-Step Solution

Answer

The absolute value of 11 is 11.

Step 1: Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering direction. Therefore, it's always a non-negative number.

Step 2: Applying Absolute Value Definition

To find the absolute value of \(11\), we consider how far \(11\) is from \(0\) on the number line. Since \(11\) is a positive number, its absolute value is \(11\) itself.

Key Concepts

Number Line

A number line is a simple yet powerful tool for understanding numbers, particularly when it comes to concepts like absolute value. Imagine a straight line that extends infinitely in both directions. At the center of this line is zero, and every point on the line represents a number.

On the right side of zero, you'll find positive numbers, and on the left side, you'll find negative numbers. Each number is equidistant from its neighboring numbers, meaning that moving one unit right or left takes you to successive whole numbers.

• The number line helps to visualize both numerical order and magnitude.
• It serves as a visual aid for understanding concepts such as distance from zero.

By plotting numbers on a number line, you can easily see how far a number is from zero, the core idea behind absolute value.

Positive Numbers

Positive numbers on a number line are found to the right of zero. They include whole numbers like 1, 2, 3, and so on, as well as fractions and decimals like 0.5 or 2.7. These numbers are greater than zero and are used in various arithmetic operations and real-life contexts.

Positive numbers have a few key characteristics:

• They indicate quantities that are greater than none, such as 5 apples, 10 dollars, etc.
• In coordinates and elevations, positive numbers usually indicate positions above or forward of a reference point.

In mathematics, positive numbers are naturally positive within the scope of understanding absolute value because they are already at a positive distance from zero. Therefore, their absolute value is the same as their intrinsic value. For instance, \( 11 \) is a positive number, and its absolute value is simply \( 11 \).

Distance From Zero

Distance from zero is the driving idea behind finding the absolute value of a number. Imagine standing at zero on a number line. The distance from zero measures how far you need to move to reach a specific number on that line, regardless of direction.

For instance, the absolute value of a number like \( 11 \) is found by determining how many units it lies from zero. Since \( 11 \) is 11 units to the right of zero, its absolute value is \( 11 \).

• Absolute value always results in a non-negative number because it's a measure of distance, not direction.
• Both positive and negative numbers can have the same absolute value if their distances from zero are equal.

For example, the number -7, which is 7 units left of zero, also has an absolute value of 7. This helps in solving problems where only the magnitude of numbers matters, not their direction.