Problem 40 - Prealgebra

Question

Find each of the following absolute values. $$|-9|$$

Step-by-Step Solution

Answer

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

Step 1: Understand Absolute Value Concept

The absolute value of a number is its distance from zero on the number line, regardless of direction. Therefore, the absolute value of any number is always non-negative.

Step 2: Identify the Number Inside

The given number is ",-9". Our task is to find its absolute value.

Step 3: Apply the Absolute Value Rule

Since absolute value concerns distance from zero, we ignore the sign of the number. The absolute value of "-9" is "+9", because 9 units lie between -9 and 0 on the number line.

Key Concepts

Number Line

A number line is a visual representation of numbers along a horizontal line, where each point on the line corresponds to a real number. The center of the number line is zero, and numbers increase positively to the right and decrease negatively to the left.
It's useful for understanding concepts like absolute value, as it visually shows how far a number is from zero. The number line features:

• Equidistant points: Each whole number is spaced evenly apart.
• Positive and Negative Indicators: Positive numbers are on the right of zero, negative numbers are on the left.
• Fractions and Decimals: Can be included for more precision.

In our example of finding the absolute value of \(-9\), we simply look at how many units \(-9\) is away from the zero point on this line, which helps us easily identify the absolute value.

Distance from Zero

When we talk about absolute value, we're really referring to the concept of distance from zero. It measures how far a number is from the zero point on the number line, without caring about which direction that number lies. For example, consider the numbers 9 and \(-9\). Both are the same distance, 9 units away from zero on the number line. Thus, both have the same absolute value: 9. Understanding this helps clarify why the absolute value is always non-negative:

• Distance can't be negative. It's only the magnitude that counts, not the direction.
• Thus, when evaluating \(|-9|\), we find it's simply 9 because \(-9\) is 9 units away from zero.

Non-Negative Numbers

Non-negative numbers are the numbers that are equal to or greater than zero. These include all the positive numbers and zero. Even though we often associate numbers with direction on a number line, the non-negative values exclude negatives.Why is the concept of non-negative numbers important in absolute value?

• Absolute value gets rid of any negativity—as shown in \(|-9| = 9\), where the result is 9.
• This is because we only care about how far \(-9\) is from zero, which turns out to be the same as how far 9 is from zero.

This means that the absolute value always corresponds to a non-negative result, reinforcing the idea that absolute values highlight size, not direction.