Problem 30 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ |-4.5| $$

Step-by-Step Solution

Answer

The absolute value of -4.5 is 4.5.

Step 1: Identify the absolute value operation

In the expression \(|-4.5|\), the bars on each side of \(-4.5\) indicate that it's an absolute value operation.

Step 2: Apply the absolute value rule

The result of the absolute value of any number is always a positive number. Hence, the absolute value of \(-4.5\) is \(4.5\).

Key Concepts

Evaluating Expressions

When we refer to evaluating expressions, we're talking about finding the value of numerical or algebraic expressions when substituting variables with concrete values, if any are present. However, with expressions that involve absolute value, like the exercise given with \( |-4.5| \), there's no need for substitution since it's already a numerical expression.

Evaluating an expression with absolute value requires us to determine the non-negative value of the number inside the absolute value bars, regardless of its original sign. In our exercise, we look at the number -4.5 that is inside the absolute value bars, and our task is to deduce its positive counterpart, which is simply 4.5.

Algebraic Operations

Algebraic operations are the mathematical procedures we use to manipulate algebraic expressions and numbers. These include basic operations like addition, subtraction, multiplication, and division, as well as more complex ones like powers, roots, and indeed, absolute values.

Understanding an absolute value as an algebraic operation is essential. It is an operation that takes a number and outputs its distance from zero on a number line. This is why absolute value is always a positive number or zero—it's a measure of magnitude, not direction. With \(|-4.5|\), we perform this operation by stripping the negative sign from -4.5, and we're left with the positive 4.5 as the output of this particular algebraic operation.

Positive Numbers

The concept of positive numbers is fundamental in mathematics. These are the numbers greater than zero, and they appear to the right of zero on the number line. They are used to represent quantities that have magnitude and are often contrasted with negative numbers, which are to the left of zero on the number line and represent quantities in the opposite direction.

In the context of the given exercise, we can see that absolute value is intimately connected with positive numbers. No matter if the input is positive or negative, the output of the absolute value function is always a positive number, as it essentially reflects the 'distance' from zero without considering direction. From the absolute value of -4.5, which is \(4.5\), we see this principle in action: the negative sign is ignored, and only the magnitude of 4.5 is considered.