Problem 67 - Elementary Algebra

Question

Simplify. $$ -||(-5) 2+42 \div 8 \| $$

Step-by-Step Solution

Answer

The simplified expression is 15.25.

Step 1: Understand the Absolute Value

Recognize that the inner absolute value symbol, \(|-5|\), finds the absolute value of -5. The absolute value of any number without its sign is 5.

Step 2: Simplify the Inner Operation

Compute the expression \(5\times2\). Multiply 5 by 2 to obtain 10.

Step 3: Divide by 8

Next, divide 42 by 8. This results in \(5.25\).

Step 4: Combine Operations within Outer Absolute Value

Calculate the expression \(||10 + 5.25|\). This becomes \(|10 + 5.25| = |15.25|\).

Step 5: Final Simplification

Recognize that the absolute value of 15.25 remains \(||15.25|| = 15.25\).

Key Concepts

Understanding Absolute Value

The concept of absolute value revolves around the idea of a number's distance from zero on the number line, regardless of its sign. By "distance," we mean how far a number is from zero, ignoring if it is positive or negative.
For instance, both 5 and -5 are 5 units away from zero, so their absolute values are both 5. In mathematical notation, we denote the absolute value with vertical bars, for example, \(|-5| = 5\).

• Absolute value only affects the sign, not the magnitude.
• It is denoted by two vertical bars, like \(|x|\).
• It always results in a non-negative value.

Recognizing and applying absolute value is crucial when solving equations where the sign can influence the outcome.

Performing Arithmetic Operations

Arithmetic operations form the foundation of algebra. These operations include addition, subtraction, multiplication, and division. Understanding how to perform these basic calculations is essential for solving more complex problems.
In our exercise:

• First, we performed multiplication: \(5 \times 2\), which equaled 10.
• Second, we carried out division: \({42 \div 8 = 5.25}\).

Every arithmetic operation follows specific properties, such as the commutative and associative properties, ensuring consistent results. Remember the order of operations, PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)), to correctly solve arithmetic expressions.

Simplification of Expressions

Simplification is the process of making an algebraic expression easier to read or understand by performing all possible operations and reducing the expression to its simplest form. The goal is to create a more manageable equation.
In the problem,

• First, we calculate \(5 \times 2\), resulting in 10, and divide 42 by 8 to get 5.25.
• Then, add these values together: \(10 + 5.25 = 15.25\).

Simplifying expressions often involves breaking down operations into smaller, easier steps, as in our example. It is a crucial skill to check if two expressions are equivalent or to solve equations efficiently. Keeping track of calculations ensures that no part of the expression is overlooked.

Working with Integers

Integers are a set of numbers that include zero, positive whole numbers, and their negative counterparts. They do not include fractions or decimals. Understanding how to work with integers is essential in various mathematical operations, especially when considering negative numbers and absolute values.
For instance, in our given exercise, we dealt with integers such as -5 and 2. The absolute value changed -5 into a positive integer, 5, demonstrating how integers can interact within absolute value calculations.

• Integers can be positive, negative, or zero.
• Operations with integers follow specific rules, such as adding negative numbers or finding absolute values.

This understanding helps students perform operations accurately and is foundational for progressing to more advanced algebraic concepts.