Problem 18 - Elementary Algebra

Question

Determine each of the values, \(-|47|\)

Step-by-Step Solution

Answer

Answer: -47

Step 1: Find the absolute value of 47

To find the absolute value of 47, we look at its distance from 0 on the number line. Since 47 is already a positive number, its absolute value remains the same: \(|47| = 47\).

Step 2: Find the negative of the absolute value

Now that we have found the absolute value of 47 to be 47 itself, we can find the negative of this value. To do this, simply add a negative sign: \(-|47| = -47\). So, \(-|47| = -47\).

Key Concepts

Negative of Absolute Value

Understanding the concept of the negative of an absolute value can be quite simple once you relate it to everyday situations. Imagine you are standing directly north from a starting point. If someone asks how far south you are, the answer would be in a negative value to indicate the opposite direction. Similarly, in mathematics, the absolute value of a number reflects its distance from zero on the number line, regardless of direction. So, the absolute value of 47, denoted as \(|47|\), is 47, because it is 47 units away from zero. To get the negative of an absolute value, you effectively reverse the direction, akin to going south instead of north. Therefore, the negative of \(|47|\) is simply -47, showing a distance of 47 units in the opposite direction from zero.

It is crucial to note that the negative of an absolute value is not the same as simply having a negative number. The absolute value symbol always yields a non-negative number because it represents distance, which is inherently non-negative. Placing a negative sign in front of the absolute value symbol, however, flips the sign of the resultant non-negative number to make it negative.

Number Line

The number line is a fundamental concept that visually represents numbers as points along a straight line. It's like a horizontal road stretching infinitely in both directions, with zero placed at the center, positive numbers to the right, and negative numbers to the left. Think of it as a ruler for measuring not just length, but also the position and direction of numbers. When we talk about absolute values, the number line becomes extremely handy because it clearly illustrates that the absolute value of a number is its 'distance' from zero.

For positive numbers, this distance is the same as the number itself because you would move to the right from zero. For negative numbers, although you head left from zero, the absolute value still represents the distance, and therefore, it's expressed as a positive number. The number line ensures a better understanding of concepts like the negative of an absolute value, as it shows that the operation changes the direction (left or right) but not the magnitude (distance) of the number.

Properties of Absolute Values

The properties of absolute values are consistent rules that help us understand and calculate the absolute value of numbers and expressions. One key property is that the absolute value of any number is always non-negative, since it's defined as the distance from zero, and distances cannot be negative. Here are a few more important properties:

• Non-negativity: As stated, \(|a| \textgreater= 0\) for any number a.
• Positive Definiteness: If \(|a| = 0\), then a must be 0.
• Symmetry: The absolute value of a number is the same as that of its negative counterpart; that is, \(|a| = |-a|\).
• Triangle Inequality: For any two numbers, the absolute value of their sum is less than or equal to the sum of their absolute values; \(|a + b| \textless= |a| + |b|\).

These properties are essential for solving various mathematical problems and proofs. They also underscore why the negative of an absolute value is straightforward: it’s simply applying a negative sign to the non-negative result of the absolute value operation.