Problem 10 - Fundamentals of Mathematics

Question

Find each value. \(-|-2|\)

Step-by-Step Solution

Answer

The value of \(-|-2|\) is \(-2\).

Step 1: Understanding Absolute Value

Absolute value represents the distance of a number from zero on a number line, without considering the direction. It is always non-negative. For any number \(x\), the absolute value is denoted as \(|x|\).

Step 2: Determine Inner Absolute Value

The expression given is \(-|-2|\). First, identify the absolute value inside the expression, which is \(-2\). Since absolute value turns a negative into a positive, \(|-2| = 2\).

Step 3: Apply Outer Negative Sign

Now, apply the negative sign outside the absolute value: \(-|2|\). As calculated, \(|2| = 2\). So, applying the negative sign results in \(- (2) = -2\).

Key Concepts

Understanding the Number Line

A number line is a simple tool used in mathematics to help us visualize numbers and their relationships. The line extends indefinitely in both directions, with zero at the center. On this line, numbers to the right of zero are positive and increase as you move further right. Conversely, numbers to the left of zero are negative and decrease as you move left.
Understanding the number line helps in grasping the concept of absolute value. Absolute value is essentially the distance of any number from zero on this number line.

• Distance is always a non-negative value.
• This ensures absolute value does not consider direction, only magnitude.

Exploring Non-negative Values

In mathematics, non-negative values are numbers that are either zero or positive. This concept is critical when dealing with absolute values because the result of an absolute value is always non-negative.

• Absolute value is defined as the distance from zero without considering direction.
• Even negative numbers become positive when their absolute values are considered, for instance, \(|-3| = 3\).
• This illustrates that absolute values transform actual negative values into easily manageable positive numbers.

Decoding Mathematical Expressions

A mathematical expression often combines numbers, variables, and operations. It represents a particular value based on the operations applied. With absolute value expressions, it's essential to first simplify the expression within the absolute value brackets.

• For example, in the expression \(-|-2|\), you initially carry out the absolute value operation inside the brackets making \(|-2| = 2\).
• Next, you process any operations that apply to the result of that absolute value calculation.
• Handling expressions stepwise aids in avoiding errors and understanding the calculations performed.

Understanding the Negative Sign

A negative sign in mathematics reverses the direction of a number on the number line. In the context of absolute values, you need to pay special attention to any negative signs outside the absolute value bars.

• When a negative sign precedes an absolute value, it affects the ultimate result after calculating the absolute value.
• As in \(-|-2|\), we solve \(|-2| = 2\) first, and then the outer negative sign makes it \(-2\).
• The negative sign ensures the direction of the number is opposite to what an absolute value alone would present.

This subtle complexity helps in understanding both absolute value and its place in broader mathematical operations.