Problem 73 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. \(-|9|\)

Step-by-Step Solution

Answer

-9

Step 1: Evaluate the Absolute Value

Find the absolute value of 9. The absolute value of any number is always positive. Therefore the absolute value of 9 is 9.

Step 2: Apply the Negative Sign

Apply the negative sign in front of the absolute value. So, \(-|9|\) becomes -9.

Key Concepts

Absolute Value

When evaluating algebraic expressions, the concept of absolute value is often encountered. The absolute value of a number is a measure of its distance from zero on the number line, without considering direction. It is always a positive value or zero. The absolute value is represented by two vertical bars surrounding the number. For example, the absolute value of both -3 and 3 is 3, denoted as \(|-3| = 3\) and \(|3| = 3\).

This concept becomes crucial when dealing with negative numbers inside absolute value symbols. It's important to remember that the outcome of the absolute value function will always be non-negative.

Application in Evaluations

The application comes into play when you encounter an expression like \(-|9|\). Initially, you must evaluate the absolute part, which results in 9, regardless of the fact that 9 is already positive. Only after finding the absolute value do you consider the negative sign prefixed to the absolute value symbol.

Negative Numbers

A negative number is a real number that is less than zero. These numbers are symbolized by a minus sign (\(-\)) in front of them. Understanding negative numbers is essential when learning about absolute values because the absolute value strips away the sign of a number and only focuses on its magnitude.

Negative Numbers in Expressions

When you work with negative numbers in algebraic expressions, such as \(-|9|\), the negative sign in front of the absolute value indicates a direction on the number line that is opposite to the positive side. If you visualize the number line, negative numbers are to the left of zero. This concept can sometimes puzzle students, especially when a negative number is displayed in an absolute value context—which ultimately yields a positive result.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (such as add, subtract, multiply, and divide). When evaluating algebraic expressions, it's essential to follow the correct order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

When an algebraic expression includes an absolute value, the contents within the absolute value bars must be evaluated first before addressing operations outside of it.

Complex Expressions

As expressions become more intricate, they can include several layers of operations wrapped with absolute values, necessitating a step-by-step approach. For example, in the expression \(-|9|\), the absolute value is a critical first step, and then the operation of applying the negative sign follows.