Problem 42 - Elementary Algebra

Question

Explain why \(|-6|=6\).

Step-by-Step Solution

Answer

The absolute value of \(-6\) is 6 because it is the distance from zero.

Step 1 - Understanding Absolute Value

The absolute value of a number is the distance of that number from zero on the number line, without considering the direction. It is always non-negative.

Step 2 - Identify the Absolute Value Operation

Recognize that \(|-6|\) represents the absolute value of \(-6\).

Step 3 - Evaluate the Absolute Value

The absolute value of \(-6\) is the distance of \(-6\) from zero. This distance is 6 units.

Step 4 - Conclusion

Since the absolute value of a negative number is always positive, \(|-6| = 6\).

Key Concepts

number line

A number line is a visual representation used in mathematics to show the position of numbers. Imagine a straight horizontal line where each point corresponds to a number. The center point, labeled as zero, is called the origin. Numbers to the right of zero are positive, and numbers to the left are negative.
Understanding the number line can help you grasp various mathematical concepts, including absolute value. For instance, the distance of \( -6 \) from zero on the number line is six units to the left. Regardless of the direction, we only consider this distance, which is how absolute values work.
Using a number line is a helpful way to visualize arithmetic operations and understand the relationships between numbers more clearly.

non-negative distance

When discussing absolute value, a crucial point to understand is 'non-negative distance.' Absolute value measures how far a number is from zero, ignoring its direction. This distance is always non-negative, meaning it’s never less than zero. For example, the distance of \( -6 \) from zero is 6 units.
Absolute value converts any negative distance into a positive figure. So, \( |-6| \) becomes 6 because distance can't be negative. It’s like measuring the length of a piece of string; it can only be a positive length, never negative.
This concept ensures that absolute value provides a simplified and universal way to discuss distances without worrying about direction.

positive numbers

Positive numbers are numbers greater than zero. They appear to the right of zero when plotted on a number line. Understanding positive numbers is essential in grasping absolute values because the absolute value of any number is always positive.
For example, even though \[ -6 \] is a negative number, its absolute value \( |-6| \) equals 6, a positive number. This is because absolute value focuses solely on distance from zero, irrespective of the number’s original sign.
When you encounter absolute values, always remember they turn any number – positive or negative – into a non-negative (or positive) number. This is crucial as it simplifies the understanding and computation of distances and magnitudes in mathematical problems.