Problem 46 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. (Lesson 2.2) $$|9|$$

Step-by-Step Solution

Answer

The absolute value of 9 is 9.

Step 1: Identify the number in the absolute value

The number inside the absolute value symbol is 9.

Step 2: Determine the distance to 0

Since 9 is 9 units away from 0 on the number line, its absolute value is 9.

Step 3: Final answer

This means the absolute value of 9, denoted as |9|, is equal to 9.

Key Concepts

Expressions

Expressions in mathematics are like secret codes that need to be decoded to understand their value. They are made up of numbers, symbols, and operators that together represent a mathematical phrase. In the context of absolute value, an expression could be something as simple as \(|9|\), where the vertical bars are not numbers, but rather symbols that enclose the number 9 inside.
\( \vert a \vert \) is the absolute value expression representing the distance of a number \(a\) from 0 on a number line.
This is important, because the idea of distance is usually associated with positive values, so the result of an absolute value expression will always be non-negative. Thus, expressions help us to communicate and solve mathematical problems in a structured way.

Number Line

Imagine a long straight line that stretches infinitely in both directions, like a road. This is your number line, which is a visual representation of numbers ordered from the left to the right. Each point on this line corresponds to a number.
The middle of the number line is zero, the center. Numbers to the right are positive, and numbers to the left are negative. To find the absolute value of a number, we use the number line to check how far the number is from zero.

• In our example, the number 9 is 9 units away from 0 on the number line.
• This means its absolute value is 9, because the absolute value measures this distance without considering direction—only the magnitude.

Understanding a number line helps you grasp the concept of absolute value intuitively, by simply counting steps from left or right to zero.

Evaluating Expressions

Evaluating expressions involves calculating their value by following mathematical rules and operations. With absolute value expressions, the goal is to determine the non-negative distance of a number from zero. You do this by ignoring any negative sign the number might have and simply counting its position on the number line.
For example, evaluating \(|9|\) is straightforward:

• Firstly, recognize that 9 is already positive, so its absolute value will remain 9.
• If it was \(-9\), we'd simply strip the negative sign, also resulting in 9, because it's 9 units away from zero in the opposite direction.

This simple operation reminds us that the absolute value cares not about direction, but about the distance—a concept that's central when evaluating such expressions.