Problem 80 - Intermediate Algebra

Question

Find the value of each expression. $$ -|-8| $$

Step-by-Step Solution

Answer

The value of \(-|-8|\) is \(-8\).

Step 1: Understand Absolute Value

Absolute value of a number represents its distance from zero on the number line, always a positive value or zero. For any negative number, the absolute value is its positive counterpart.

Step 2: Find the Absolute Value

Calculate the absolute value of \(-8\).The absolute value is denoted as \(|-8|\), which equals 8 because removing the negative sign gives its positive form.

Step 3: Evaluate The Expression

Substitute the absolute value back into the original expression. Replace \(|-8|\) with 8, giving us the expression \(-8\).

Step 4: Simplify the Expression

Evaluate \(-8\) by simply negating the positive number 8.So, the expression \(-|-8| = -8\).

Key Concepts

Negative Numbers

Negative numbers represent values less than zero. They are visually recognizable on a number line by being positioned to the left of zero.

These numbers are an important aspect of mathematics, as they allow us to represent losses, deficits, or temperatures below freezing, among other things.

• They are typically symbolized with a minus sign (-) before the number, such as -1, -5, or -8.
• Mathematically, adding two negative numbers results in a more negative number, while multiplying or dividing two negative numbers yields a positive product or quotient.
• Subtraction involving negative numbers can be a bit tricky, as subtracting a negative is equivalent to adding the positive version of that number

By understanding how negative numbers operate, students can more easily evaluate expressions that include them, ensuring precise calculations and better handling of situations and phenomena involving negative values.

Number Line

The number line is a fundamental visual tool used in mathematics to represent numbers and their relationships to each other. It is essentially a straight line with defined points marked at regular intervals, each representing a number.

Here’s how it aids in understanding mathematical concepts:

• The center of the number line is typically zero, which acts as a neutral point separating positive and negative numbers.
• To the right of zero, numbers are positive and increase in value the further right you go.
• To the left of zero, numbers are negative and decrease in value the further left you go.
• The absolute value of a number is its distance from zero, irrespective of its direction (left or right). This is why |-8| is equal to 8, as it shows the distance without considering direction.
• It gives a clear, visual understanding of various operations, such as addition or subtraction, especially involving negative numbers.

Incorporating number lines when learning about numerical expressions can demystify troublesome concepts and empower students to solve problems with greater confidence and comprehension.

Expression Evaluation

Expression evaluation is the process of determining the numerical value of a mathematical expression. It requires systematically following specific rules and conventions to achieve an accurate result. In the context of absolute values and negative numbers, this becomes even more crucial:

• Start by evaluating operations inside any absolute value brackets first, to understand the 'positive form' of that segment.
• Absolute values take precedence because they reflect necessary core adjustments to the terms within.
• Substitute the simplified form of the absolute value back into the expression.
• Apply standard arithmetic operations, following the order of operations (parentheses, exponents, multiplication and division, addition and subtraction, commonly abbreviated as PEMDAS).
• Verify your solution by checking it against the number line, as it confirms both directionality and magnitude.

By methodically evaluating expressions, especially those with elements like negative numbers and absolute values, students develop accuracy in their calculations and gain a deeper insight into how numerical elements interact.