Problem 103 - Elementary Algebra

Question

Simplify. $$ |-33| $$

Step-by-Step Solution

Answer

33

Step 1: Understand the Absolute Value

The absolute value of a number is the non-negative value of that number without regard to its sign. It represents the distance of a number from zero on the number line.

Step 2: Apply the Absolute Value Operation

Given the number is -33, its absolute value is 33 because absolute value removes the negative sign.

Step 3: Write the Simplified Expression

After calculating the absolute value, we write the simplified expression, which is 33.

Key Concepts

Simplification of Absolute Values

When dealing with absolute values, simplification means finding the non-negative version of a number. The absolute value, denoted by vertical bars like \(|-33|\), essentially tells us how far a number is from zero on the number line. For the problem \(|-33|\), we simplify by removing the negative sign, resulting in \(33\).
This process strips away any negative signs to show only distance, which is why you always end up with a positive or zero value after simplification.

• Absolute value alters negative numbers to positive.
• Non-negative values remain unchanged.

Simplifying expressions, especially when absolute values are involved, lets us focus on the magnitude rather than the direction of a number.

Using the Number Line for Visualization

A number line is a helpful visual tool that clarifies the concept of absolute value. Imagine a long line with zero at the center, numbers increasing positively to the right, and decreasing negatively to the left.
When you consider the number \(-33\), it would be placed 33 units to the left of zero. The absolute value measures the number's distance from zero, which is always positive, hence \(|-33| = 33|\).

• Zero remains a neutral point.
• Distance is measured in positive units from zero.

The number line helps to illustrate why absolute values are always positive by showing distance in a straightforward manner.

Understanding Negative Numbers in Absolute Values

Negative numbers on the number line are values located to the left of zero. They tell us about values that are smaller than zero. However, when we calculate absolute values, these negative signs are no longer considered important.

If \(-33\) is the number in question, think of \(-33\) as being 33 units left from zero. Here, the absolute value ignores the sign and only counts these units, making it \(|-33| = 33|\).

• Absolute values make negatives behave as their positive counterparts.
• This concept is crucial for measuring distance without direction.

So, the role of negative numbers within absolute values is just to determine the initial direction, which isn't considered in the final absolute value.