Problem 60 - Algebra and Trigonometry

Question

Rewrite each expression without absolute value bars. $$\|-5|-|-||13$$

Step-by-Step Solution

Answer

The expression \(|-5|-|-|13\) without absolute values is 8.

Step 1: Compute first absolute value

Take the first absolute value, which is |-5|. Because -5 is negative, the absolute value of -5 is its positive value, or 5.

Step 2: Compute second absolute value

The second absolute value to compute was |13|. Since 13 is positive, the absolute value remains the same, which is 13.

Step 3: Subtract the computed absolute values

After computing the absolute values, we need to subtract the result from Step 1 from the result in Step 2. In other words, perform |13|-5, which gives us 8.

Key Concepts

Positive and Negative Numbers

Understanding positive and negative numbers is essential for working with absolute values. In mathematics, the number line helps us visualize these numbers. Positive numbers are to the right of zero and include values like 1, 2, 3, and so on. Negative numbers are to the left of zero, like -1, -2, -3, etc.

The absolute value of a number, represented by two vertical bars (e.g., \(|-5|\) or \(|13|\)), is its distance from zero on the number line. Absolute values are always positive because distance cannot be negative. For example, the absolute value of \(-5\) is \(5\), since it is 5 units away from zero.

When we deal with expressions involving absolute values, we often simplify by considering whether each number inside the bars is positive or negative. This simplification helps us focus on the numerical value without concern for its sign.

Mathematical Expressions

Mathematical expressions combine numbers, variables, and operation symbols to represent a value. In our example, we dealt with \(\||-5|-|13|\). Here, the absolute value bars are part of the expression, which signifies that we need to handle each part carefully.

To simplify such expressions, follow these steps:

• Identify each sub-expression that requires separate calculation. In our case, these are \(|-5|\) and \(|13|\).
• Compute the absolute values independently.
• Finally, combine the results using the given operations, such as subtraction or addition.

By breaking down the expression and computing each part, we ensure accuracy and avoid confusion.

Arithmetic Operations

Arithmetic operations form the foundation of solving mathematical expressions. Addition, subtraction, multiplication, and division are the primary operations.

In the exercise, the operation involved is subtraction. After computing the absolute values, \(|13| = 13\) and \(|-5| = 5\), the next step is to perform \(13 - 5\). This operation means taking away a smaller value from a larger value, resulting in a positive number: \(8\).

Whenever performing arithmetic operations involving absolute values, it’s crucial to:

• Compute absolute values first to eliminate the bars.
• Carefully execute the operation according to the mathematical rules.
• Double-check the work to ensure accuracy.

This method ensures that the expression is simplified correctly, giving a clear and final answer.