Problem 112 - Elementary Algebra

Question

Simplify. $$ -|-8| $$

Step-by-Step Solution

Answer

The simplified expression is -8.

Step 1: Understand the Absolute Value

Absolute value refers to the distance of a number from zero without considering the direction on the number line. For example, \(-8\) has an absolute value of 8 because it is 8 units away from zero.

Step 2: Simplify the Absolute Value

Remove the absolute value sign by converting \(-8\) into 8. This gives us \(|-8| = 8\).

Step 3: Apply the Negative Sign

Multiply the result of the absolute value by -1, as indicated by the problem. Thus, we have \(-(|-8|) = -8\).

Key Concepts

Understanding Negative Numbers

Negative numbers are an interesting concept as they represent values less than zero. They are typically indicated by a minus sign (-) before the number. Imagine a thermometer measuring temperatures below freezing: these readings would be negative numbers. Since they are less than zero, they appear to the left of zero on the number line.

Negative numbers have some unique properties. For instance, multiplying two negative numbers results in a positive number. Also, when you add a negative number to a positive one, you are essentially subtracting the absolute value of the negative number from the positive number. This can sometimes lead to surprising results, like how adding a negative to a smaller positive will give a negative result.

• Negative imes Negative = Positive
• Positive imes Negative = Negative
• Negative + Positive = Depends on which has a larger absolute value

Exploring the Number Line

The number line is a valuable tool in understanding numbers, both positive and negative. It is a straight line where each point corresponds to a number. Zero is commonly placed in the center, with positive numbers extending to the right and negative numbers extending to the left.

Understanding the number line can help in visualizing basic math operations like addition and subtraction. For instance, when subtracting a positive number, you move left on the number line, i.e., toward more negative values. Conversely, adding a positive number will take you right, toward higher values.

The beauty of the number line is its ability to represent the magnitude of numbers (their size or absolute value) as distances from zero. This is why when dealing with absolute values, you look at the distance from zero, ignoring which direction you are on the line.

The Art of Simplification

Simplification in mathematics involves making expressions or equations easier to understand and work with. It often means breaking down complex expressions into simpler forms. In the case of absolute values, simplification involves removing the absolute value signs and adjusting the expression accordingly.

In the given exercise, the simplification process requires you to evaluate the absolute value first. For instance, the absolute value of \(-8\) is \(|-8| = 8\) since it is 8 units away from zero. After finding the absolute value, it is necessary to consider the remaining operations—in this case, applying a negative sign to the result. This step-by-step breakdown ensures that even complicated expressions become manageable.