Problem 72 - Elementary and Intermediate Algebra

Question

Find each absolute value. $$|4.3|$$

Step-by-Step Solution

Answer

|4.3| = 4.3

Step 1: Understand Absolute Value

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

Step 2: Identify the Number

Look at the number inside the absolute value sign. In this case, it is 4.3.

Step 3: Determine the Absolute Value

Since 4.3 is already a positive number, its absolute value remains the same. Therefore, \( |4.3| = 4.3 \)

Key Concepts

distance from zero

When we talk about absolute value, a crucial idea to understand is the 'distance from zero.' Imagine you are standing on a number line. The distance from zero is simply how far you have to walk to reach the number you are looking at, without considering the direction. Whether the number is to the left of zero (negative) or to the right of zero (positive), the distance is always counted as a positive number. For instance, the distance from 4.3 to zero is 4.3 units. The same applies to -4.3; its distance from zero would also be 4.3 units. Therefore, the absolute value of any number is its distance from zero, making \(|4.3| = 4.3 \).

number line

The number line is a visual tool used in mathematics to represent numbers in a straight line. Each point on this line corresponds to a unique number. Let's break it down further:

• **Zero** sits right at the center, acting as our home base.
• **Positive numbers** line up to the right of zero.
• **Negative numbers** stretch to the left of zero.

So when we talk about absolute value, we are thinking about how far a number is from this central point of zero. If we plot the number 4.3 on this line, it will be to the right of zero. To find the absolute value, all you need to consider is how many steps or units you must move from zero to reach this number, which is exactly 4.3 in this case. Therefore, \(|4.3|=4.3 \).

non-negative number

An important aspect of absolute value is that it is always a non-negative number. This means the result can either be positive or zero, but it will never be negative. The term 'non-negative' is simpler than you might think:

• 'Non' means 'not.' So, non-negative means 'not negative.'
• Therefore, non-negative numbers include all positive numbers, as well as zero.

If we look at the example \( |4.3| \), the absolute value here is 4.3, which is a positive number. No matter whether the original number is positive or negative, the absolute value will turn it into a non-negative result. This helps us to easily compare distances without worrying about directions. Hence, the solution \(|4.3|=4.3 \) perfectly showcases that absolute values are always non-negative.