Problem 3 - Precalculus

Question

Evaluate each expression. $$|-6|$$

Step-by-Step Solution

Answer

The absolute value of $$-6$$ is $$6$$ .

Step 1: Understanding Absolute Value

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

Step 2: Compute the Absolute Value

To find the absolute value of $$-6$$ , we look at how far $$-6$$ is from zero. Since $$-6$$ is 6 units away from zero on the number line, $$|-6| = 6$$ .

Key Concepts

Number Line

A number line is a visual representation of numbers along a straight line. This tool helps visualize how numbers relate to each other and to zero. On a number line, numbers increase as you move to the right and decrease to the left. Zero is the central point.

• Positive numbers are to the right of zero.
• Negative numbers are to the left of zero.
• Each point on the line represents a unique number.

Using the number line helps in understanding the concept of absolute value because it shows exactly how far a number is from zero. For example, if you look at data{\(-6\)}, it is located to the left of zero, 6 units away. Visualization on the number line helps confirm that the absolute value of \(-6\) is 6.

Distance from Zero

The distance from zero determines the absolute value of a number. Absolute value focuses solely on how far a number is from zero, disregarding whether the number is negative or positive.
To calculate this distance, simply count the units between your number and zero on the number line. This measures the magnitude of the number, which is always considered a positive distance.
For instance:

• Number \(-6\): 6 units away from zero, so absolute value is 6.
• Similarly, \(+6\) is also 6 units away, making its absolute value 6.

This process highlights that absolute value is all about distance, not direction.

Non-Negative Value

A non-negative value is any number that is either positive or zero. Absolute values are always non-negative because they represent the magnitude or size of a number in terms of distance from zero.
No matter how negative a number is, its absolute value is expressed as a non-negative number. This is because absolute value ignores whether the number is positive or negative, focusing only on its size.
In calculations:

• Absolute value of \(-6\) is 6, a non-negative number.
• Similarly, absolute value of \(0\) is 0, which is not negative.
• Absolute value of any positive number remains unchanged.

This principle is crucial, especially in mathematical contexts where distances or magnitudes are discussed without the need for direction.