Problem 12 - Precalculus

Question

Evaluate each expression. $$||-8|-|-9||$$

Step-by-Step Solution

Answer

The expression evaluates to 1.

Step 1: Understand the Problem

The expression to evaluate is \(||-8| - |-9||\). The absolute value symbol \(|x|\) represents the distance of a number from zero, which means it is always non-negative. We need to evaluate the nested absolute values step by step.

Step 2: Evaluate Inner Absolute Values

First, calculate the absolute values inside the expression: 1. Evaluate \(|-8|\), which is 8 because the absolute value turns a negative number into its positive counterpart. 2. Evaluate \(|-9|\), which is 9 because of the same reason.

Step 3: Subtract the Inner Results

Now subtract the results from Step 2: Calculate \(8 - 9\), which equals \(-1\).

Step 4: Evaluate the Outer Absolute Value

Finally, evaluate the absolute value of the result from Step 3: Calculate \(||-1||\), which is 1 since the absolute value of \(-1\) is 1.

Key Concepts

Evaluating Expressions

When evaluating expressions involving absolute values, each part must be handled strategically. The absolute value, noted with vertical bars (| |), signifies how far a number is from zero on the number line. It transforms negative numbers into positive ones. For instance, in the expression \[||-8| - |-9||\], we dissect the problem in sections. Consider each absolute value separately, acting as a two-layer problem-solving journey. Always begin with the inner expressions, resolve, and work outward. Simplifying methodically in this manner leads to a clear understanding of how values interact within expressions.

Nested Absolute Values

Nested absolute values might look complicated, but with practice, they become manageable. When you see nested absolute values, like \[||-8| - |-9||\], think of them as layers. Each set of absolute values—inner and outer—carries its task.

• **Inner Absolute Values**: Focus on computing the absolute values of the numbers, where \(|-8| = 8\) and \(|-9| = 9\).
• **Subtract Inner Values**: Once simplified to 8 and 9, subtract to find the difference, resulting in \(8 - 9 = -1\).
  By addressing inner values and progressing outward, each layer resolves into simpler forms until the entire expression is evaluated. The absolute value's role brings structure by ensuring all results reflect positive, real-number distances.

Precalculus Problem Solving

Precalculus includes solving problems using concepts like absolute values. When approaching expressions with multiple layers, strategic step-by-step execution is essential. Situations like evaluating \[||-8| - |-9||\] teach the core principles:

• Recognizing how absolute values transform negative inputs.
• Systematically solving from the innermost part of the expression to the outermost.
  These exercises sharpen mathematical thinking by requiring careful breakdowns of tasks. Such processes are foundational for complex analysis, as they teach us to approach and simplify tasks meticulously. Emphasis on strategy and understanding aids in problem-solving, preparing students for more advanced concepts in mathematics.