Problem 71 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. \(|2|\)

Step-by-Step Solution

Answer

The value of |2| is 2.

Step 1: Understanding the Absolute Value concept

The absolute value of a real number reveals its distance from zero or its magnitude without considering its sign. It is symbolized by two straight lines enclosing the number. This means the absolute value will always be non-negative. In mathematical terms, for a real number x, we have, |x| = x if x >= 0, and |x| = -x if x < 0.

Step 2: Applying the Absolute Value Definition

In this exercise, we are asked to evaluate the absolute value of 2, which is written as |2|. According to the definition of absolute value, if the number is positive or zero, |number| is the number itself. Since 2 is a positive number, |2| equals 2.

Key Concepts

Evaluate Expressions

Evaluating expressions is a fundamental aspect of algebra. It involves finding the value of a mathematical expression when certain values are substituted for variables. When there are no variables, as in the case of evaluating the absolute value of the number 2, the process is straightforward but still requires understanding the underlying concepts. To properly evaluate an expression like \(|2|\), one must apply specific rules or definitions that apply to the mathematical operation in question—in this case, the operation is finding the absolute value.

An easy way to picture this is to think of an expression as a set of instructions. Just as a recipe tells you how to combine ingredients to make a dish, evaluating an expression tells you how to combine numbers and operations to find a result. With absolute value, the 'recipe' is simple: determine the distance of the number from zero on a number line, without concern for direction.

Real Numbers

Real numbers encompass a broad range of values that include integers, fractions, decimals, and irrational numbers. They can be positive, negative, or zero. In the context of absolute value, which requires the input of a real number, understanding the nature of these numbers is critical.

For instance, real numbers can be visualized on a number line, which extends infinitely in both positive and negative directions, with zero at the center. The importance here is that the absolute value of a real number is its distance from zero on this line, and by distance, we mean a non-negative value. Whether you are dealing with the square root of 2, which is an irrational number, or -5, an integer, the absolute value will always give you a real number that is equal to or greater than zero, reflecting the idea of 'magnitude' without 'direction'.

It's also vital to note that when evaluating expressions that contain real numbers, the context can often dictate whether the number itself or its absolute value is required.

Mathematical Terms

Understanding mathematical terms is akin to learning the vocabulary of a new language. It is much easier to grasp complex concepts when you are familiar with the definitions of terms like 'absolute value'. In our exercise, the term 'absolute value' can be defined as the distance of a number from zero on the number line without considering its sign. This term is critical because it denotes a quantity's size regardless of its direction from zero.

One important trait of the absolute value is that it is always a non-negative number. This is why positive numbers stay the same and negative numbers become their positive counterparts when placed within absolute value symbols. A fluent understanding of mathematical terms will not only assist you in the evaluation of expressions but will also provide a firm foundation for solving more complex mathematical problems.

Furthermore, grasping these terms helps in correctly interpreting mathematical statements and instructions, ensuring that you follow the correct procedures to arrive at a valid solution.