Problem 80 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ -0.3-0.2 $$

Step-by-Step Solution

Answer

The result of the operation is \(-0.5\).

Step 1: Understand the operation

The exercise requires us to perform the subtraction operation between two negative decimal numbers: \(-0.3 - 0.2\). Subtraction can often be rephrased as adding the opposite, so this is the same as adding \(-0.3 + (-0.2)\).

Step 2: Combine the numbers

Combining the numbers involves adding their absolute values and retaining the negative sign since both numbers are negative. The operation will look like this:\(-0.3 + (-0.2) = -(0.3 + 0.2)\).

Step 3: Add the absolute values

Add the absolute values of 0.3 and 0.2. \(0.3 + 0.2 = 0.5\).

Step 4: Apply the negative sign

Since both the original numbers are negative, apply the negative sign to the combined result. This gives us:\(-0.5\).

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers less than zero and are usually represented with a minus sign (-) before them. They hold a crucial role in mathematics, particularly when dealing with real-world problems such as temperature changes or financial calculations that reflect debt. Think of negative numbers as the opposite of positive numbers. While positive numbers might represent gaining or increasing value, negative numbers often represent a loss or decrease.

When working with negative numbers in operations like subtraction, it’s important to remember that subtracting a negative number is like adding its positive counterpart. For example, \(-0.3 - 0.2\) can also be considered \(-0.3 + (-0.2)\), where you combine the values and retain the negative sign. This concept is vital as it helps simplify calculations, ensuring consistent results.

The Role of Absolute Value

Absolute value represents the non-negative value of a number without regard to its sign. It measures the distance of a number from zero on the number line, regardless of direction.

In mathematical terms, the absolute value of a number \(x\) is denoted as \(|x|\). For example, the absolute value of both \(+0.3\) and \(-0.3\) is \(0.3\), since they are both \(0.3\) units away from zero.

When performing operations involving negative numbers, such as subtracting or adding negative decimals, absolute values come into play to simplify the process. In the exercise \(-0.3 - 0.2\), we focus on the absolute values (\(0.3\) and \(0.2\)) to help combine the numbers before attaching the negative sign for the final result. By adding these absolute values, \(0.3 + 0.2 = 0.5\), and applying the negative sign, you get \(-0.5\). Understanding absolute values makes handling calculations involving negative numbers much easier.

Addition of Decimals

Adding decimals might appear tricky, but the process is straightforward with attention to detail. Here's what to keep in mind when adding decimals:

• Align the numbers by their decimal points. This ensures each digit is in the correct place value.
• Add placeholders (like zeroes) if necessary, to ensure both numbers have the same number of decimal places.
• Consider the sign of each number—negatives require extra attention.

In the example \(-0.3 + (-0.2)\), aligning the decimal points results directly into adding the absolute values \(0.3\) and \(0.2\). This gives a simpler perspective on the math and ensures that the final addition is accurate. As shown in the step-by-step solution, adding \(0.3\) and \(0.2\) results in \(0.5\). Finally, since the numbers being added are both negative, the result is also negative, hence \(-0.5\). Understanding how to carefully add decimals and monitor positive and negative signs ensures accuracy and confidence in your math skills.