Problem 7 - Elementary Algebra

Question

Simplify the following problems. $$ |-3| $$

Step-by-Step Solution

Answer

Answer: The absolute value of -3 is 3.

Step 1: Understanding Absolute Value

The absolute value of a number, written as |x|, is the distance between that number and 0 on the number line. The result is always non-negative, as distance cannot be negative. The absolute value of x can be defined as: If x >= 0, then |x| = x If x < 0, then |x| = -x

Step 2: Applying the Definition of Absolute Value to |-3|

Now, we will find the absolute value of -3. Since -3 is less than 0, we will use the second part of the definition: |-3| = -(-3)

Step 3: Simplify the result

Finally, let's simplify the result: |-3| = 3 Therefore, the absolute value of -3 is 3.

Key Concepts

Understanding the Number Line

The number line is a visual representation of numbers in a straight, horizontal format. Each point on the line corresponds to a real number, with zero located in the center. Negative numbers extend to the left, and positive numbers to the right.

Absolute value is intimately connected to the number line because it essentially measures the "distance" of a number from zero. For instance, when we look at -3 on the number line, we see it is 3 units away from zero. Despite being on the negative side, distance is always positive, which is why we say the absolute value of -3 is 3.

• Absolute value tells us how far a number is from zero on the number line.
• Distances on the number line are always non-negative.
• Positive numbers and their negative counterparts have the same absolute value.

The Nature of Non-negative Values

Non-negative numbers are those that are either positive or zero. They do not include any negative numbers. When talking about absolute value, we are always referring to non-negative outcomes. This is because absolute value is all about the distance from zero, and distance cannot be negative.

Whenever we calculate the absolute value of a number, our result will be a non-negative number. For example:

• |3| = 3, which is non-negative.
• |-3| = 3, which is non-negative as well.
• |0| = 0, which is also non-negative.

This concept assures us that irrespective of the direction on the number line, the measure of how far off zero we are will never be a negative figure.

Simplification Process in Absolute Value

Simplifying expressions involving absolute value is an essential skill. The simplification process is straightforward but requires understanding the definition of absolute value first.

Let's simplify |-3| as an exercise. By definition, the absolute value \(|-x|\) is equal to -(-x) when x is negative, because we're essentially "flipping" the negative sign. Therefore, |-3| is simplified as follows:

1. Identify the value: -3
2. Recognize that -3 is negative.
3. Apply the formula: |-3| = -(-3)
4. Simplify: -(-3) = 3

This results in |-3| being equal to 3, emphasizing that the simplification process involves converting any negative number into its positive counterpart when determining absolute value. Understanding this process makes it easier to handle more complex equations involving absolute values.