Problem 12 - Elementary Algebra

Question

Determine each of the values, |14|

Step-by-Step Solution

Answer

Answer: The value of |14| is 14.

Step 1: Identify the number within the absolute value notation

The number within the absolute value notation is 14.

Step 2: Determine the absolute value of the number

Since 14 is a positive number, its absolute value remains the same. So, the absolute value of 14 is 14. The value of |14| is 14.

Key Concepts

Positive Numbers

Positive numbers are those that are greater than zero. They sit on the right side of zero on a number line. In everyday life, positive numbers can represent quantities like the amount of money you have or the number of apples in a basket. They are straightforward because they don’t have a negative sign in front of them. For example:

• 3, 12, and 100 are all positive numbers.
• These numbers have their sign naturally assumed as positive, even if a positive sign isn't marked explicitly (i.e., +3).

Specific properties of positive numbers make them unique. Unlike their negative counterparts, adding or multiplying positive numbers always yields a positive number. Dividing is a little different, as the quotient could be a positive or negative number depending on the context. But when we consider an absolute value, which we'll discuss soon, the nature of the number as being 'positive' or 'negative' matters a lot.

Absolute Value Notation

Absolute value is a concept you’ll come across in mathematics frequently. The notation for absolute value uses vertical bars or lines, like this: \(|x|\). These bars tell us to focus on the size or magnitude of the number, ignoring its positive or negative sign.

When you see a number in absolute value notation, such as \(|14|\), it's an instruction to find out how far the number is from zero on the number line. This distance is always expressed as a non-negative number. For the example \(|14|\), the distance from zero is simply 14.

The concept is used because sometimes only the magnitude of a number matters, not whether it's considered negative or positive. You'll often find absolute value used in situations where direction is irrelevant, like measuring time or distance. Knowing how to interpret this notation can clarify problems requiring a deeper analysis of numbers.

Determining Absolute Value

Determining the absolute value is a simple process. The key is understanding that absolute value strips away any negative sign, if present. This means that:

• If the number is positive, as we've seen with 14, the absolute value remains the same. Thus, \(|14| = 14\).
• If the number is zero, its absolute value is also zero, because zero is neither negative nor positive.
• If the number is negative, say \(-7\), it becomes positive, \(|-7| = 7\).

The step-by-step approach to determining absolute value involves checking the number inside the bars:
1. If the number is positive or zero, the absolute value is that number itself.
2. If the number is negative, convert it by removing the negative sign, thus finding its positive counterpart.This method helps in recognizing the distance of any number from zero without considering its direction, ensuring clarity in various mathematical and real-world contexts.