Problem 99 - Prealgebra and Introductory Algebra

Question

Evaluate. $$|-x|, \text { for } x=8$$

Step-by-Step Solution

Answer

The value of \(|-x|\) for \(x = 8\) is 8.

Step 1: Recognize the Absolute Value Function

The problem involves an absolute value function, denoted as \(|-x|\). It represents the non-negative value of \(x\). Whether \(x\) is positive, negative, or zero, \(|-x|\) will always be a non-negative number.

Step 2: Substitute the Given Value

The value given for \(x\) is 8. Substitute \(x = 8\) into the absolute value function.

Step 3: Evaluate the Absolute Value

For \(x = 8\), the absolute value \(|-x|\) becomes \(|-8|\). The absolute value of -8 is 8, as the absolute value function essentially removes the negative sign.

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero. They play an important role in mathematics and can be found on the left side of the number line. Positive numbers have a plus sign, while negative numbers are indicated by a minus sign. You can think of them like temperatures below freezing or elevations below sea level.

• In math operations, adding a negative number results in subtraction.
• Multiplying two negative numbers yields a positive result.
• Meanwhile, multiplying a negative by a positive number results in a negative.

Understanding how negative numbers behave is crucial for solving problems involving absolute values because it often involves removing the negative sign when finding the magnitude. Mounting the concept of negative numbers in your mind is like building a foundation that helps with substitution, evaluation, and dealing with the absolute value function.

Substitution in Mathematics

Substitution is a method used in mathematics to replace a variable with a given number or expression. It acts as a tool for solving equations or evaluating expressions. In our exercise, we begin by substituting the value 8 for the variable \(x\).

• Substitute the given number directly into the expression.
• Replace every occurrence of the variable with this number.
• Always ensure that you use parentheses to clarify order in complex expressions.

By substituting \(x = 8\) into \(|-x|\), we convert it into \(|-8|\). This transformation simpliflies the absolute value evaluation, as we're working towards finding the value of the expression with a specific numerical replacement.

The Process of Evaluation

Evaluation refers to the calculation that determines the numerical outcome or value of an expression. After substitution, the expression \(|-8|\) needs to be evaluated properly.

• Begin by taking note of the absolute value function.
• Observe any mathematical operation or sign changes involved.
• Close out by simplifying the numerical expression to reach a final value.

For the expression \(|-8|\), evaluation involves computing the absolute value of \(-8\), which gives us 8. This process requires understanding the effect of the absolute value function on negative numbers, as it converts them to their positive counterpart.

Grasping the Absolute Value Function

The absolute value function is key when dealing with negative numbers. Denoted by vertical bars \(|x|\), it represents the distance of a number from zero on the number line. Thus, the absolute value is always a non-negative number.

• For any positive number, the absolute value remains the same.
• For zero, the absolute value is zero.
• For any negative number, the absolute value removes the negative sign, resulting in a positive number.

In our exercise, the expression \(|-x|\) implies taking the absolute value of a negative number. No matter what value \(x\) holds, \(|-x|\) means stripping away the negative sign to reveal the positive magnitude of the number. This inherent ability to turn negatives into positives makes the absolute value function essential in calculating distances, norms, and differences in various mathematical scenarios.