Problem 69 - Prealgebra

Question

Simplify each of the following. $$-|-8|$$

Step-by-Step Solution

Answer

The simplified expression is -8.

Step 1: Evaluate the Absolute Value

The expression is -|-8|. We start by evaluating the absolute value |-8|, which represents the distance of -8 from 0 on the number line. Regardless of the sign, the absolute value turns the number into its positive equivalent.

Step 2: Simplify the Absolute Value

Now, simplify the absolute value: |-8| = 8. Since the absolute value of any number is always non-negative, |-8| is equal to 8.

Step 3: Apply the Negative Sign

With the absolute value calculated in Step 2, substitute it into the original expression: -|-8| becomes -8. The negative sign outside the absolute value indicates that we take the negative of the absolute value.

Key Concepts

Absolute Value

When working with \'absolute value\', you are considering the distance of a number from zero on the number line. This is a key concept because it is always a non-negative number. **Think of absolute value as a way of simplifying a number to its basic, positive form**, reflecting its magnitude without regard to its sign. So, for any number \( x \), the absolute value \( |x| \) is simply \( x \) if \( x \) is positive, and \(-x\) if \( x \) is negative.

For instance, consider the number \(-8\). Its absolute value is \(|-8| = 8\) because regardless of being negative, it is 8 units away from zero. This ability to ignore the sign makes absolute value particularly useful in prealgebra for solving equations and inequalities.

Negative Numbers

Negative numbers can be puzzling because they represent values less than zero. On a number line, they appear to the left of zero. These numbers are important in many math situations, like when discussing temperatures below freezing or debts.

• **Negative signs** indicate direction on a number line.
• If you see "-" before a number, it means you owe or are below zero.

When dealing with expressions like \(-|-8|\), it’s important to first simplify what’s inside the absolute value before applying any external negative signs. Here, the expression prompts you to first convert \(-8\) inside the absolute value brackets to its positive form, 8, and then apply the negative sign outside, resulting in \(-8\).

This logical flow aligns with the order of operations, emphasizing the necessity to resolve absolute values before external operations, crucial for managing negative numbers in computations.

Simplification

Simplification is the process of making an expression easier to understand and work with. This is often done by performing all calculations so that the expression is left in its simplest form. In regards to expressions involving combination operations, such as absolute values and negatives, you apply these steps sequentially.

To **simplify** the expression \(-|-8|\), follow these steps:

• Calculate the absolute value: \(|-8| = 8\).
• Apply the negative sign: \(-|-8| = -8\).

These steps show why following the correct order of operations is crucial for simplification in prealgebra. Each transformation reflects a clear change to the expression, avoiding confusion and correctly solving mathematical problems. This clarity is a cornerstone of mathematical problem-solving, ensuring that calculations are logical and precise.