Problem 29 - Fundamentals of Mathematics

Question

Determine each of the values. $$ -(-|-6|) $$

Step-by-Step Solution

Answer

The value is 6.

Step 1: Understand the Absolute Value

Absolute value refers to the distance a number is from zero on the number line, regardless of direction. It is always positive. Hence, \(|-6|\) simplifies to \(6\).

Step 2: Handle the Inner Negative Sign

Evaluate the expression \(-|-6|\). By replacing \(|-6|\) with \(6\) from Step 1, you get \(-6\).

Step 3: Simplify the Outer Negative Sign

Now simplify \(-(-6)\). The double negative flips the sign of \(6\), resulting in a positive \(6\).

Key Concepts

Negative Numbers

Negative numbers are those that are less than zero and are represented with a minus sign (-). They can be a bit tricky because they introduce the idea of direction on the number line. **For example:** -6 means six units to the left of zero on the number line. In daily life, negative numbers can represent things like debt or temperatures below zero.
Understanding how negative numbers interact with other numbers is essential in math. **Key Points to Remember:**

• Negative multiplied or divided by another negative makes a positive. So, -(-6) becomes +6.
• Adding a negative number is the same as subtracting its positive counterpart.
• Negative numbers open up a world of real-world applications in finance, science, and beyond.

By mastering negative numbers, you gain a powerful tool for solving countless math problems!

Number Line

The number line is a very useful tool that helps us visualize numbers and their relationships to one another. It stretches infinitely in both directions but is typically shown with zero in the center. Positive numbers go to the right, while negative numbers stretch to the left. This visual representation is crucial to understanding absolute value and operations involving negatives.
When you're finding an absolute value, you actually determine how far a number is from zero. **For example:** On the number line, both -6 and 6 are six units away from zero. Therefore, they have the same absolute value. Benefits of using a number line include:

• Quickly identifying which numbers are greater or smaller.
• Easily seeing the effects of adding and subtracting negative numbers.
• Visual aid to comprehend the properties of absolute values.

Mastering the number line concept enables you to grasp more complex mathematical ideas.

Simplifying Expressions

Simplifying expressions means reducing them to a more concise or simpler form without changing their value. It often involves operations like removing parentheses, combining like terms, and performing any calculations needed. When dealing with absolute values and negative signs, simplification follows certain arithmetic rules.
In the expression \[-(-|-6|)\], start by solving the absolute value, which gives 6. Then handle the double negative, which simplifies to simply 6 again. Why simplify?

• Creates more manageable equations.
• Highlights the core components of a problem for analysis.
• Efficiently communicates the results without unnecessary complexity.

Understanding these simplification principles can enhance problem-solving skills and improve efficiency in calculations!