Problem 75 - Intermediate Algebra

Question

Evaluate each expression. See Example \(9 .\) $$ -2|4-8| $$

Step-by-Step Solution

Answer

The value of the expression is \(-8\).

Step 1: Evaluate the expression inside the absolute value

The expression inside the absolute value is \(4 - 8\). Subtract \(8\) from \(4\) to get \(-4\).

Step 2: Apply the absolute value

The absolute value of a number is its distance from zero on the number line, regardless of direction. The absolute value of \(-4\) is \(4\), so \(|4-8| = 4\).

Step 3: Multiply by -2

Now multiply the result from Step 2 by \(-2\): \(-2 \times 4 = -8\).

Key Concepts

Understanding Absolute Value

The concept of absolute value revolves around the distance of a number from zero on a number line, leaving aside any directional aspects. For any given number, regardless of whether it is positive or negative, the absolute value is always non-negative. This makes absolute value a crucial concept in mathematics, especially when dealing with expressions and equations.

• The absolute value symbol is represented by vertical bars, such as \( |x| \).
• For any positive number \(x\), the absolute value is simply \(x\): \( |x| = x \).
• For any negative number \(-x\), the absolute value is \(x\): \( |-x| = x \).

The absolute value basically asks, "How far is this number from zero?" Hence, whether a number is \(-4\) or \(4\), the absolute distance is \(4\). Therefore, the absolute value of \( -4 \) is \( 4 \), because it lies four units away from zero on the number line.

Mastering Integer Operations

Integer operations are foundational in evaluating expressions, particularly when you're required to perform addition, subtraction, multiplication, or division with whole numbers.

• Addition & Subtraction: When you add or subtract integers, you follow the sign rules. Adding a negative number is the same as subtracting the corresponding positive, and vice versa.
• Multiplication & Division: The product or quotient of two integers will be positive if the numbers have the same sign, and negative if they have different signs.

In expressions like \( 4 - 8 \), understanding integer operations is key. Subtracting \(8\) from \(4\) results in \( -4 \), as you effectively add \( -8 \) to \(4\). Mastery of these operations paves the way for smoothly navigating through more complex mathematical problems.

Step-by-Step Solutions for Expression Evaluation

A systematic approach, or step-by-step solution, is crucial for solving mathematical problems correctly and efficiently. Let's break down the approach for evaluating expressions involving absolute value and integers.

• Step 1: Simplify inside expressions first. If the expression involves combinations of operations, handle the operations inside absolute value bars first, as was done with \( 4 - 8 = -4 \).
• Step 2: Apply absolute value. Calculate the absolute value of your result from Step 1, as absolute value cannot be negative. So \( |-4| = 4 \).
• Step 3: Consider multiplication or further operations. When the result from the absolute value operation is achieved, apply any further operations outside the absolute value bars, such as multiplying by \( -2 \), resulting in \( -8 \).

This step-by-step strategy assures accuracy and lessens errors, as each calculation is directly based on the preceding result. Taking it one step at a time avoids misinterpretations and builds a clear path from problem to solution.