Problem 15 - Elementary Algebra

Question

Determine each of the values, |-16|

Step-by-Step Solution

Answer

Answer: 16

Step 1: Identify the number inside the absolute value expression

In this exercise, the number inside the absolute value expression is -16.

Step 2: Evaluate the absolute value expression

To find the absolute value of -16, we simply take the distance from -16 to 0 on a number line, disregarding the negative sign. This distance is 16. Therefore, |-16| = 16.

Key Concepts

Understanding the Number Line

A number line is a visual representation of numbers arranged in a straight line. It helps you see where numbers are located relative to each other. Numbers increase in value as you move right and decrease as you go left. On a number line, zero is typically placed in the center.

Here’s why number lines are useful:

• They provide a clear visual of order among numbers, making it easier to compare them.
• You can use them to perform basic arithmetic, such as addition and subtraction.
• They are essential for understanding concepts like absolute values, because they show distance from zero.

For absolute value, it’s about finding how far a number is from zero. Thus, even if you have a negative number like -16, the distance on the number line from -16 to 0 is identical to that from 16 to 0.

Calculating Distance on a Number Line

In mathematics, distance refers to how far apart two numbers or points are on a number line. But what does this actually mean in terms of calculation?

To determine this distance, you follow these simple steps:

• Identify the two points between which you want to measure the distance.
• Consider only the absolute difference between these points, ignoring negative signs.
• For example, the distance from -16 to 0 is 16 units, just as from 0 to 16.

Distance always measures as a positive value, representing how many steps you would need to take to go from one point to the other without regard for direction.

Why Only Positive Values?

The notion of distance is purely about magnitude, not direction. That’s why it's always positive. If you think about traveling or moving from one place to another, distance doesn’t care if you’re moving backward or forward.

Grasping Negative Numbers

Negative numbers might seem daunting, but they are just numbers like any other, with values less than zero. On a number line, negative numbers lie to the left of zero, and each step to the left decreases the number’s value.

Understanding negative numbers is crucial for several reasons:

• They’re used in real-life contexts, such as temperatures below freezing or debts.
• They are essential when solving equations and understanding absolute values.
• Learning how to manage these numbers will enhance your mathematical problem-solving abilities.

When working with absolute values, you "strip away" the negative sign because the aim is to find the distance from zero. So, the absolute value of -16 acknowledges only 16 steps from zero, discarding the direction indicated by the negative sign.