Problem 51 - College Algebra

Question

Rewrite expression without absolute value bars. \(|300|\)

Step-by-Step Solution

Answer

The expression \(|300|\) without absolute bars is 300.

Step 1: Understand Absolute Values

An absolute value of a number is its distance from zero regardless of the direction - whether positive or negative. It simply measures how far a number is from zero. For any real number x, the absolute value is denoted as \(|x|\).

Step 2: Apply the Absolute Value Concept

Here, we are to find the absolute value of 300, which is already a positive number. Thus its absolute value remains the same. So the absolute value of 300, \(|300|\), is 300 itself.

Key Concepts

Real Numbers

Real numbers are part of the foundational building blocks in mathematics. They include all numbers that are not imaginary, encompassing both positive and negative numbers, as well as zero.

• These numbers can either be rational or irrational.
• Rational numbers are those that can be expressed as a fraction, such as \(<\frac{1}{2}\) or -2. Whole numbers and integers also fall under this category.
• Irrational numbers cannot be written as a fraction; a common example is \(\pi\) or the square root of 2.

Most measurements and calculations in the real world involve real numbers, making them an essential topic in mathematics. Understanding real numbers allows us to comprehend other concepts such as absolute value, which directly deals with real numbers only.

Distance from Zero

When we talk about absolute values, we are essentially discussing a number's distance from zero on the number line. This concept is vital in understanding absolute values.

• An absolute value, represented as \(|x|\), indicates how far the number is from zero, regardless of the direction on the number line.
• For example, both -5 and 5 have an absolute value of 5, because they are both five units away from zero.
• This gives insight into why the absolute value of any positive number or zero is the number itself, while the absolute value of a negative number is its positive counterpart.

Distance from zero helps simplify expressions and solve equations, such as when calculating the absolute difference between numbers.

Positive and Negative Numbers

Positive and negative numbers are integral to the concept of absolute value. They reflect a number's position relative to zero on the number line.

• Positive numbers are greater than zero and are found to the right of zero on the number line.
• Negative numbers, however, are less than zero and are located to the left of zero.
• Each number's absolute value turns it into its distance from zero, eliminating the negative sign if present.

Understanding positive and negative numbers aids in grasping how absolute values unify them into a single measure of distance. It underscores that absolute value is a measure and not a reflection of whether a number is positive or negative.