Problem 74 - Introductory and Intermediate Algebra for College Students 4th

Question

Find each absolute value. $$|-9|$$

Step-by-Step Solution

Answer

The absolute value of \(-9\) is \(9\).

Step 1: Understand Absolute Value

The absolute value of a number is its distance from 0 on the number line, regardless of direction. It is always non-negative: \(|a| \geq 0\).

Step 2: Apply the Definition

We evaluate the absolute value of the given expression. Since the absolute value removes any negative sign, we take the non-negative value.

Step 3: State the Result

The absolute value of \(-9\) is \(9\).

Key Concepts

Algebra and Absolute Value

In algebra, absolute value is one of the essential concepts that students encounter. It represents the distance a number is from zero on a number line, irrespective of direction. This means that the absolute value of a number is always non-negative. For instance, when you come across a problem such as finding the absolute value of \( -9 \), the primary step in solving this is to recognize that the absolute value symbol, denoted by two vertical bars \( | \cdot | \), is asking for just the magnitude of the number.

To better understand, envision that you owe \$9 — the distance from having no debt (zero) is \$9, which is true whether you owe it or possess it as cash. Therefore, the algebraic operation involved here simplifies to removing the sign of the number, making \( |-9| = 9 \). In this way, algebra teaches us to focus on the size or quantity without concern for the negative or positive sign.

Understanding Distance from Zero

The term 'distance from zero' is another way of describing the absolute value, and it's a fundamental principle that can be applied in various contexts, including real-life scenarios. Distance, by definition, is always positive, as it doesn't make sense to talk about negative distances. For example, the distance between two points, or in this case from a point to zero, can not be negative.

When you look at the problem \( |-9| \), you are essentially being asked, 'How far is -9 from zero?' To answer this, you don't need to consider which direction you have to travel on the number line; you simply count the units. This is why the distance from zero for both \( +9 \) and \( -9 \) is \( 9 \). Simply put, whether you move 9 steps forward or 9 steps backward from zero, your distance travelled is 9 steps.

The Number Line as a Tool for Absolute Value

The number line is an incredibly visual tool that helps in understanding many concepts in algebra, especially the absolute value. It's akin to a ruler that measures the distance of numbers from zero. On a number line, zero is typically placed in the middle, with positive numbers to the right and negative numbers to the left.

Regarding the problem \( |-9| \), if you were to plot -9 on a number line, it would appear 9 units to the left of zero. To find the absolute value, you'd look at how many units you'd have to move to get back to zero, ignoring the direction (left or right).

Visualizing Absolute Value

As you begin to visualize this, you can see that moving from -9 to 0 covers 9 units. Similarly, moving from 9 to 0 also covers 9 units. That’s the power of using a number line—it concretely shows that the absolute value of -9 is equal to the positive number 9. By practicing with a number line, the concept of absolute value becomes transparent and intuitive.