Problem 12 - Algebra 2 and Trigonometry

Question

Find the value of each given expression. \(|8|-|-2|-|2|\)

Step-by-Step Solution

Answer

The value of the expression is 6.

Step 1: Understand Absolute Values

Absolute values measure the distance of a number from zero on the number line and are always non-negative. The absolute value of a number is written as \(|x|\). For example, \(|8| = 8|\) and \(|-2| = 2|\).

Step 2: Evaluate Inner Absolute Values

Evaluate \(|-2|\) from the expression. Since \(-2\) is 2 units away from zero, we have \(|-2| = 2|\).

Step 3: Evaluate Outer Absolute Values

Now evaluate \(|2|\) from the result of the previous step. Since \(2\) is already positive, \(|2| = 2|\).

Step 4: Evaluate Full Expression

Substitute the values obtained: \(|8|-|-2|-|2| = 8 - 2 = 6|\).

Key Concepts

integer operations

Integer operations are fundamental in mathematics. They involve basic arithmetic operations such as addition, subtraction, multiplication, and division with whole numbers, called integers. Integers can be positive, negative, or zero.
Understanding these operations helps in solving math problems efficiently. Here's a brief overview of each:

• Addition: Adding positive integers involves moving to the right on the number line, while adding negative integers moves you left.
• Subtraction: Subtracting an integer is the same as adding its opposite. For example, subtracting a positive integer is like adding a negative integer.
• Multiplication: Multiplying two integers with different signs results in a negative product, while two integers with the same sign result in a positive product.
• Division: Similar to multiplication, division follows the same sign rules for determining a positive or negative quotient.

Integer operations are crucial for solving expressions that involve absolute values, as these operations are needed to simplify or calculate the results once absolute values are evaluated.

step-by-step solution

Solving math problems with a step-by-step approach ensures clarity and accuracy. This method breaks down a problem into manageable parts, making it easier to tackle complex expressions.
For instance, when dealing with absolute values like the exercise offering the expression offering the expression offering offered:

• Step 1: Understand the definition. Recognize what absolute values represent. They reflect the distance from zero, meaning they're always non-negative.
• Step 2: Evaluate inner components. Start with the innermost absolute values, as they’ll contribute to simplifying outer expressions.
• Step 3: Compute outer values. Once inner values are determined, move outward to resolve the next layers of absolute values.
• Step 4: Provide a solution. With each evaluation, substitute back into the main expression to reach a final result.

This structured approach cultivates a deeper understanding, leading to successful problem-solving skills.

distance from zero

The concept of distance from zero is pivotal when understanding absolute values. Absolute values are essentially the non-negative distance a number holds from zero on a number line.
In mathematical terms, the absolute value of a number \( x \) is represented as \( |x| \).

• For positive numbers, the absolute value is the number itself, such as \( |8| = 8 \).
• For negative numbers, it’s the positive counterpart of the number, for instance, \( |-2| = 2 \).
• The distance from zero ensures that results using absolute values are simplified consistently, even with negative values, as seen with \( |2| = 2 \).

By understanding the distance from zero concept, you can confidently solve problems involving absolute values, ensuring you always achieve non-negative results.