Problem 26 - Elementary Algebra

Question

Determine each of the values, \(-|-19|\)

Step-by-Step Solution

Answer

Answer: -19

Step 1: Find the absolute value of -19

The absolute value of a number is its distance from 0 on a number line, without considering direction. So, the absolute value of any negative number will be its positive counterpart. The absolute value of \(-19\) is denoted as \(|-19|\) and can be calculated as follows: \[ |-19| = 19 \]

Step 2: Take the negative of the absolute value

Now, we have the absolute value of \(-19\) which is 19. We are asked to find the negative of this value. This can be calculated simply by adding a negative sign to 19: \[ -|-19| = -19 \] The value of \(-|-19|\) is \(-19\).

Key Concepts

Negative Numbers

When we talk about negative numbers, we are referring to numbers that are less than zero. These are often marked with a minus sign (-). They are located to the left of zero on a number line. Negative numbers can be thought of as a representation of owing something rather than having it. They are important in many real-world situations, such as indicating temperatures below freezing or when you are in debt in financial contexts.

One interesting property of negative numbers is when you multiply them by a negative number, the result turns out to be a positive number. For instance, \(-2 \times -3 = 6\). Understanding negative numbers helps in dealing with concepts of loss or deficiency compared to positive numbers.

Number Line

A number line is a straight, horizontal line that is used to represent numbers in increasing order from left to right. It's a very useful tool for visualizing numbers and understanding their relationships.

On a number line, zero is the central point. Positive numbers are positioned to the right of zero, while negative numbers are placed to the left. This setup allows for an easy way to look at how numbers compare to each other.

• Moving to the right on the number line means increasing the value.
• Moving to the left means decreasing the value.

Understanding how to use a number line can help in performing operations like addition and subtraction of numbers.

Distance from Zero

The distance from zero, also known as the absolute value, refers to how far a number is from zero on a number line, without considering direction. This is always a non-negative number because distance is positive.

For example, the absolute value of both \( -19 \) and \( 19 \) is \( 19 \) because both are 19 units away from zero.

• If a number is negative, its absolute value will be its positive counterpart.
• If a number is already positive or zero, the absolute value remains unchanged.

Understanding the concept of distance from zero is essential as it is widely used in different areas of mathematics, like solving equations and understanding norms and metrics in more advanced math.