Problem 10 - Algebra 2 and Trigonometry

Question

Find the value of each given expression. \(|8+(10-18)|\)

Step-by-Step Solution

Answer

The value is 16.

Step 1: Evaluate Inside the Parentheses

First, solve the expression inside the parentheses: \(10 - 18\). This gives \( -8\). So, the expression becomes \(8 + |-8|\).

Step 2: Apply Absolute Value

The absolute value of a number is the distance between that number and zero on a number line, always expressed as a non-negative number. So, \(|-8| = 8\). The expression is now \(8 + 8\).

Step 3: Perform the Addition

Now add the numbers: \(8 + 8 = 16\).

Key Concepts

Absolute Value

Absolute value is a critical concept in algebra, particularly when dealing with expressions or equations involving negative numbers. It is denoted by vertical bars, such as \(|x|\), and signifies the distance of a number from zero on a number line. Because distance is inherently non-negative, absolute value is always positive unless the number itself is zero, in which case the absolute value is zero.

For example, consider the number -8. On the number line, -8 is 8 units away from 0. Thus, the absolute value of -8 is 8, written as \(|-8| = 8\).

In practical terms, when evaluating the absolute value, you can think of it as removing any negative signs from within the absolute value bars. This makes calculations easier, avoiding errors when dealing with complex expressions that involve both positive and negative numbers.

Order of Operations

The order of operations is a set of rules used in mathematics to clarify which procedures to perform first in a given mathematical expression. This concept ensures consistency in solving math problems, avoiding confusion and errors.

The order can be remembered by the acronym PEMDAS, which stands for:

• **P**arentheses
• **E**xponents (powers and roots, for example, squares and square roots)
• **M**ultiplication and **D**ivision (from left to right)
• **A**ddition and **S**ubtraction (from left to right)

Following this order, you begin by solving expressions inside parentheses, then move on to exponents. After that, execute multiplication and division as they appear from left to right, and finally, perform addition and subtraction in their left-to-right order.

In the given problem, you first handled the expression within the parentheses \((10 - 18)\), then applied absolute value, and finally added the results. This is a classic example of applying the order of operations correctly to reach the final answer.

Parentheses in Math

Parentheses are used in math to group terms and clarify the order in which operations should be performed. They have the highest priority in the order of operations, meaning expressions within parentheses should be solved first. This concept is essential for solving complex expressions and ensuring that calculations yield the correct result.

Consider the expression given in the exercise: \(8 + (10 - 18)\). To solve this, you must first evaluate the expression within the parentheses. The subtraction inside gives \(-8\). By resolving parentheses first, you eliminate any ambiguity in how terms and operations should be processed.

This precedence of parentheses helps organize calculations and is especially important in more complex algebraic expressions, ensuring that each part of an expression is tackled in the correct sequence. After handling the operations within parentheses, you can then move on to dealing with other operations outside, following the order laid out in PEMDAS.