Problem 20 - Elementary Algebra

Question

Determine each of the values, |-9|

Step-by-Step Solution

Answer

Answer: The absolute value of |-9| is 9.

Step 1: Identify the number inside the absolute value brackets

In this case, the number inside the absolute value brackets is -9.

Step 2: Determine the distance from 0 without the sign

Since absolute value only cares about the distance from 0, we can ignore the negative sign for this step. The distance between -9 and 0 will be the same as the distance between 9 and 0.

Step 3: Write the absolute value

The absolute value of |-9| is equal to the distance between 9 and 0, which is 9. The absolute value of |-9| is 9.

Key Concepts

Distance From Zero

Understanding the concept of distance from zero is crucial when dealing with absolute values. In mathematics, the term 'distance' usually refers to the positive difference between two points along the number line, regardless of direction. This concept is a fundamental part of the absolute value, which essentially asks the question, 'How far is this number from zero?'.

When working with absolute values, such as in the example of \(|-9|\), the distance from zero is 9 units on the number line. It doesn't matter whether the number is to the left or to the right of zero; the absolute value symbol, represented by | |, strips away any notion of direction and focuses solely on the magnitude of the number in relation to zero. Essentially, it measures how 'long' it is from zero, not caring about the 'direction' it's in.

Negative Numbers

Negative numbers are numbers less than zero and they appear to the left of zero on the number line. They are often used to represent a deficit or a value decrease in everyday situations, like temperature below freezing or debt. In our example, \(-9\) is a negative number, located 9 units to the left of zero on the number line.

When dealing with negative numbers, especially in the context of absolute values, it's important to remember they are simply a representation of position. The absolute value function transforms this number to its positive counterpart, effectively 'mirroring' it to the same distance on the right of zero. The purpose of this transformation is to quantify the size of numbers without considering the aspect of direction or position on the number line. This approach can particularly be useful when calculating differences or when considering magnitudes in elementary algebra.

Elementary Algebra

Elementary algebra is the branch of mathematics that deals with the general properties of numbers and the rules for manipulating these properties. It's a foundational piece of any mathematical education and a vital tool in understanding and solving problems related to absolute values. In the action of finding the absolute value of a number, such as in our example with \(|-9|\), the principles of elementary algebra come into play.

If you consider the absolute value as a function, which takes a number and returns its distance from zero, you are applying algebraic thinking. You're recognizing that whether a number is positive or negative, the function (in this case, the absolute value) treats both in a similar way, resulting in a positive outcome. This ties back into the idea of distance from zero: algebra teaches us that no matter the input's sign, the outcome reflects a non-negative value, which is the underlying principle when evaluating absolute value expressions.