Problem 17 - Elementary Algebra

Question

Determine each of the values, \(-|12|\)

Step-by-Step Solution

Answer

Answer: The value of \(-|12|\) is -12.

Step 1: Find the absolute value of 12

To find the absolute value of 12, we simply look at its distance from 0 on the number line. Since 12 is a positive number, its distance from 0 is just 12 itself. Therefore, \(|12| = 12\).

Step 2: Multiply the absolute value by -1

We are trying to find the value of \(-|12|\). Since we know that \(|12| = 12\), we can substitute this into the expression: \[-(|12|) = -(12).\] Now we just need to multiply -1 by 12: \[-(12) = -12.\]

Step 3: Final Answer

The value of \(-|12|\) is -12.

Key Concepts

Number Line

The number line is a fundamental concept in mathematics that provides a visual representation of numbers. It is usually drawn as a straight, horizontal line with intervals marked at equal distances. The center of the line is marked by zero (0), which is known as the origin. Numbers to the right of zero are positive, and numbers to the left of zero are negative. For instance, when we plot the number 12 on the number line, we move twelve units to the right of zero.

Understanding the number line is crucial when dealing with absolute values. The absolute value of a number is its distance from zero on the number line, regardless of direction. By visualizing the position of the number 12, we can see that its absolute value, denoted by \( |12| \), is indeed 12, since it lies 12 units away from the origin on the number line.

Negative Numbers

Negative numbers are found on the left side of zero on a number line and represent values less than zero. They are indicated by a minus sign (-) before the number. Negative numbers are used in various contexts, such as temperature below zero, below sea level depths, or financial debts.

In the given exercise, after finding the absolute value of 12, we then apply the negative sign to it, effectively flipping its position on the number line from 12 units to the right of zero to 12 units to the left. Consequently, the operation \( -|12| \) yields \( -12 \) because it represents a point that's 12 units in the negative direction from the origin.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. In the context of absolute values and negative numbers, algebraic expressions allow us to describe operations applied to these numbers, even when they are far from reach or abstract.

For example, the expression \( -|12| \) involves taking the absolute value of 12 and then multiplying by -1. This can be interpreted as two distinct steps in the world of algebra that tell a story: first, find how far 12 is from zero (perform \( |12| \) to get 12), and then change its direction (multiply by -1 to get \( -12 \) ). This algebraic expression simplifies to a single negative number, showing the power of algebra to compactly express complex numerical relationships.