Problem 77 - Intermediate Algebra

Question

Find the value of each expression. $$ |-5.9| $$

Step-by-Step Solution

Answer

The value of \(|-5.9|\) is 5.9.

Step 1: Understanding Absolute Value

The notation \(|-5.9|\) represents the absolute value of \(-5.9\). Absolute value measures the distance of a number from zero on the number line, regardless of direction. Thus, it is always non-negative.

Step 2: Applying Absolute Value

For any negative number \(x\), \(|x| = -x\). Therefore, the absolute value of \(-5.9\) is \(-(-5.9)\). This is because the absolute value removes the negative sign, giving us the positive of the number.

Step 3: Calculating the Result

Calculate \(-(-5.9)\), which simplifies to \(5.9\) since the two negatives cancel each other out.

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero, represented with a minus sign, such as -5, -9.3, or -5.9 as seen in the exercise. They are an integral part of mathematics, allowing us to express values less than nothing. These types of numbers are typically found:

• In temperature readings below freezing levels
• In financial contexts, like debt or loss
• In measuring elevations below sea level

Understanding negative numbers is key to grasping other mathematical concepts, such as absolute value.
It is important to note that when dealing with negative numbers, arithmetic operations, like addition and multiplication, follow specific rules that differ from positive numbers. For instance, multiplying two negative numbers yields a positive result, while adding two negative numbers results in a more negative number.

Number Line

A number line is a visual representation of numbers in a straight line.
It is a simple and effective tool used to understand the basics of mathematics. In a typical number line:

• Zero is located at the center
• Positive numbers extend to the right of zero
• Negative numbers extend to the left of zero

Imagine the number line as a ruler, where each point corresponds to a number.
This visualization helps in understanding operations such as addition, subtraction, and notably, the concept of absolute value. With the number line, you can easily see how far a number, whether positive or negative, is from zero; this is essentially its absolute value.

Distance from Zero

The distance from zero is often referred to as the absolute value of a number.
In simpler terms, it's about how far a number is from zero on the number line, regardless of which direction it lies—left or right. For example, both -5.9 and 5.9 have the same distance from zero, which is 5.9.
This is because absolute value ignores the direction (negative or positive) and focuses purely on the distance:

• For positive numbers, absolute value is the number itself
• For negative numbers, it’s the positive version of the number

Understanding the distance from zero is crucial for solving equations and comprehending real-world applications where negative values can occur, such as owing money or moving below ground level.