Problem 79 - Intermediate Algebra

Question

Find the value of each expression. $$ -|-6| $$

Step-by-Step Solution

Answer

The value of the expression is -6.

Step 1: Simplify the Absolute Value

Start by evaluating the absolute value expression. The absolute value of a number is its distance from zero on the number line, without considering which direction from zero the number lies. Calculate \(|-6| = 6\).

Step 2: Apply the Negative Sign

Now, apply the negative sign in front of the absolute value expression from the original problem. Since we have -\(|-6|\) and \(|-6| = 6\), this becomes \(-6\). The negative sign outside the absolute value affects the entire value obtained from the absolute operation.

Key Concepts

Negative Numbers

Negative numbers are an essential part of the number system. They are the numbers that appear to the left of zero on the number line.
Negative numbers are represented by a minus (-) sign in front of the number, like -1, -2, -6, etc. Negative numbers are less than zero and they decrease in value as they move further to the left on the number line. If you think about temperature, being below zero is an excellent practical example of negative numbers.
Negative numbers have some interesting properties, especially when combined with operations like addition, subtraction, multiplication, or division. When added to a positive number, the negative value decreases the total sum. When two negative numbers are multiplied, the result is positive.
It's crucial to recognize negative signs in mathematical expressions because they affect the entire value of the expression, as seen in the given problem. Here, applying the negative sign to the absolute value result inverts its value.

Number Line

A number line is a straight horizontal line that visually represents numbers. The center point of the number line is labeled zero. To the right of zero, numbers increase positively, and to the left, they become more negative.
Number lines are essential for understanding absolute values. The absolute value of a number is its distance from zero on the number line.
Therefore, whether the number is positive or negative, its absolute value is the same magnitude. For example, on a number line, the distance between -6 and zero is 6 units. Hence, |-6| = 6.
The representation on a number line helps in understanding how accounts balance out positive and negative values. To visualize, think of the positive direction as steps forward and negatives as steps backward. When considering an expression involving absolute values, the idea of how far a number is from zero, regardless of direction, is central.

Simplifying Expressions

Simplifying expressions is the process of breaking down complex expressions into simpler forms, making calculations more manageable. This often involves:

• Removing parentheses
• Combining like terms
• Applying mathematical operations that simplify the calculation

In the given problem, simplifying starts with calculating the absolute value. Once the absolute value \(|-6|\) is simplified to 6, the next step involves other operations, such as applying a negative sign.
This is a classic example of simplification where understanding the individual components is key. Simplifying expressions means ensuring the expression is in its most general form, with no further simplifications needed.