Problem 47 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. (Lesson 2.2) $$|-32|$$

Step-by-Step Solution

Answer

The value of \(|-32|\) is 32.

Step 1: Understand the meaning of absolute value

The absolute value of a number represents its distance from zero on a number line. It always gives a non-negative result.

Step 2: Evaluate

Since the number given, -32, is located 32 units from 0, the absolute value of -32 is 32. It means regardless of the direction from 0, the distance is always counted as positive.

Key Concepts

Number Line

A number line is a visual representation that helps us understand the absolute value, among other mathematical concepts. Imagine a straight line with equally spaced marks where each mark represents a different integer. Zero usually sits right at the center of this line. Numbers to the right of zero are positive, while those on the left are negative.

Why is the number line important? It provides a clear way to visualize the "distance" elements involved with absolute value. To find the absolute value of a number, simply look at how far that number is from zero on the number line. For instance, both -32 and 32 are located 32 marks away from zero. Remember, on the number line, the direction (left or right) doesn't matter when calculating absolute value—only the distance counts.

Distance from Zero

When we talk about the absolute value, we emphasize the concept of distance from zero. On the number line, this "distance" is represented by the number of units a particular number is away from zero.

For example, if you consider the number -32, it sits 32 units away from zero. This distance is the same as positive 32. This is why the absolute value of -32 is calculated as 32.

• Distance does not depend on direction: Whether the number is negative or positive, the distance from zero is calculated the same way, ensuring absolute values are non-negative.
• Always think of it as steps on the number line, with zero as the starting point.

Non-negative Result

One fundamental feature of the absolute value is that the result is always non-negative, no matter the original number. This is because absolute value measures distance, and distance can't be negative.

Returning to our example of -32, while the number itself is negative, it is still 32 units away from zero. Hence, the absolute value is 32, a non-negative number, reflecting a principle in math: distances are always shown as positive values.

• Non-negative results eliminate confusion about the direction on the number line.
• This concept simplifies mathematical operations, emphasizing the magnitude or size, rather than location in relation to zero.

Understanding these characteristics helps in grasping why absolute value expressions uniformly produce a non-negative outcome.