Problem 42 - Algebra 1

Question

Evaluate the expression. $$|-4|$$

Step-by-Step Solution

Answer

The absolute value of -4 is 4.

Step 1: Understand Absolute Value

The absolute value of a number is its distance from zero on the number line. More formally, if a is a real number, then the absolute value of a, written as |a|, is defined as follows: If a is positive or zero, then |a| = a. If a is negative, then |a| = -a. Therefore the result of |-4| is always positive because absolute value makes any number nonnegative.

Step 2: Evaluate the Absolute Value

The absolute value of |-4| is 4. The sign inside the absolute value notation doesn't affect the outcome, it's always a positive number or zero.

Key Concepts

Understanding Real Numbers

Real numbers are a broad category of numbers that includes both rational and irrational numbers. They form the complete number line that stretches infinitely in both directions. If you think about any number you can place on this line, it is a real number.
For instance:

• Whole numbers like 0, 1, 2, 3,...
• Fractions like \(\frac{1}{2}\), \(\frac{3}{4}\)
• Decimals like 0.75, or even repeating decimals like 0.333...
• Irrational numbers like \(\pi\) and \(\sqrt{2}\)

Real numbers are crucial in understanding the absolute value because any real number can have an absolute value, representing its distance from zero.

Navigating the Number Line

A number line is a visual representation of real numbers, drawn as a straight horizontal line. Zero is typically placed at the center, with positive numbers extending to the right and negative numbers extending to the left.
Each point on the number line corresponds to a real number. This tool is essential for understanding how absolute values work. The absolute value represents the position's distance from zero on this line.
For example, the number -4 is four units away from zero. We don't consider direction when determining absolute value, just the distance. Therefore, both -4 and 4 have an absolute value of 4.

Differentiating Positive and Negative Numbers

Positive and negative numbers are basic components of real numbers. Positive numbers are greater than zero, and they are often found on the right side of the number line. They include numbers like 1, 2, 3, and so on.
Negative numbers, on the other hand, are less than zero and can be found on the left side of the number line. These include -1, -2, -3, etc.
Understanding positive and negative numbers is crucial when dealing with absolute values. When you find the absolute value of a negative number, you disregard the negative sign to express its distance from zero positively. Thus, \(|-4|\) results in 4, regardless of the initial negative sign.