Problem 1 - Precalculus

Question

Evaluate each expression. $$|3|$$

Step-by-Step Solution

Answer

The value is 3.

Step 1: Identify the Absolute Value Expression

The expression given is \(|3|\). We need to evaluate the absolute value of the number 3.

Step 2: Understand Absolute Value

The absolute value of a number, represented by vertical bars \(|x|\), is the non-negative value of that number. It is the distance from zero on the number line, regardless of direction.

Step 3: Apply Absolute Value Rule

According to the rule of absolute value, \(|x| = x\) if \(x\) is positive or zero. Since 3 is positive, \(|3| = 3\).

Step 4: Write the Result

The calculation is complete. We find \(|3| = 3\).

Key Concepts

Number Line

The number line is a visual representation that helps us understand and compare numbers. It looks like a straight horizontal line with an origin point, typically at zero. Positive numbers are located to the right of the zero, while negative numbers stretch out to the left. Every number on this line has a position or coordinate.

The interesting part about the number line is its use in real-world applications, such as temperature scales, financial debts, and sports scores. It serves as a fundamental tool for understanding concepts like absolute value. Absolute value measures the distance of a number from zero without considering which direction it is. On the number line, regardless of whether you move left or right from zero, the distance is always non-negative, which helps clarify why absolute values - Ignore the sign of a number - Help us find distance between numbers - Are always non-negative Use a number line as a handy tool for visualizing how numbers compare and relate to each other in terms of absolute distance.

Positive Numbers

Positive numbers are numbers greater than zero. On the number line, they are found to the right of zero. These numbers include whole numbers like 1, 2, 3, and fractions or decimals like 0.5, 1.75. When evaluating an expression like \[|3|\],we can say that 3 is a positive number because it is located right of zero on the number line.

Positive numbers express quantities like wealth (money you receive) or scores (points you've earned), where having more is favorable. In mathematical rules and interpretations:- Positive numbers retain their value under absolute value- They model quantities like height, age, and speed- Manipulating them in operations is straightforwardThey are foundational in mathematics, representing gain or growth everywhere they appear.

Non-Negative Value

Understanding non-negative value is essential in working with absolute values. A non-negative value is any number that is zero or positive. This can include zero itself, such as in the equation \[|0| = 0\],or any positive number like 3, where \[|3| = 3\].
In essence:- Non-negative includes zero and all positive numbers- Absolute value transforms any input into a non-negative value- It's crucial when calculating distances or magnitudesWhen dealing with equations or real-life problems, knowing that calculations yield non-negative results can prevent errors and ensure correct interpretations. This principle guarantees that the absolute value always provides a direction-independent measure, helping simplify and standardize calculations no matter the input.