Problem 40 - Algebra 1

Question

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

Step-by-Step Solution

Answer

-4.5

Step 1: Recognize the Absolute Value

The absolute value of a number is the non-negative value of a number without regard to its sign. It is denoted by the symbols '| |'. In this case, we have \( |-4.5| \).

Step 2: Calculate the Absolute Value

The absolute value of -4.5 is 4.5, because when we disregard the sign of -4.5, we are left with 4.5. So, \( |-4.5| \) = 4.5.

Step 3: Apply the Negation

The entire expression is negated which means that we apply a negative sign to the absolute value calculated in step 2. Thus, \( -|-4.5| \) = -4.5.

Key Concepts

Evaluating Expressions

Evaluating expressions is a fundamental concept in algebra and involves finding the value of a mathematical statement. When we have an expression like

• \(-|-4.5|\)

it involves performing operations in a specific order. First, we should clear any mathematical functions such as absolute values before addressing other operations like negation. This helps us simplify complex expressions step by step into more manageable numbers.

The priority of operations follows the order indicated by mathematical conventions, often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication/Division, and Addition/Subtraction. However, when evaluating expressions with absolute values, absolute values often serve like parentheses, thereby taking priority during simplification.

By following this standard procedure in evaluating expressions, you ensure clarity and prevent errors, allowing for a more accurate mathematical calculation.

Understanding Negation

Negation in mathematics refers to the operation of taking the opposite of a number. While learning algebra, it's crucial to understand how this affects expressions.

In the example given:

• \(-|-4.5|\)

we notice that after we evaluate the absolute value of \(-4.5\) (which is \(4.5\)), a negation symbol \(-\) is then applied. This turns \(4.5\) into \(-4.5\). The purpose of negation is to change the sign of the number it is applied to, effectively reflecting the number across zero on the number line.

Key points about negation:

• It affects all real numbers and is often symbolized by the "-" sign.
• For example, the negation of \(3\) is \(-3\), and the negation of \(-7\) is \(7\).
• Negation switches positive numbers to negative and negative numbers to positive.

Understanding negation helps in solving equations and inequalities, where changing the side of a number can lead to an entirely different outcome. Negation is a simple yet powerful operation that maintains balance in mathematical expressions.

Essentials of Algebra 1 Concepts

Algebra 1 is often the first exposure students have to algebraic expressions and operations. Core topics covered include understanding expressions, like evaluating and simplifying them, and basic operations such as addition, subtraction, multiplication, and division involving both numbers and variables.

The expression

• \(-|-4.5|\)

is an example of combining these basic concepts:

• Absolute values, as seen with \(|-4.5|\).
• Negation is applied after the absolute value.
• Emphasis on order of operations to solve expressions properly.

All these foundational elements give students the skills they need to tackle more complex algebraic problems later. Building a strong understanding of these basics is essential because it allows students to confidently approach equations and functions, which are the backbone of higher-level mathematics.

Algebra 1 also introduces students to solving equations and understanding functions, creating a solid base for further mathematical study. Mastering these concepts is crucial for future success in math.