Problem 34 - Beginning and Intermediate Algebra

Question

Add. See Examples I through 7. $$ -6.7+(-7.6) $$

Step-by-Step Solution

Answer

The sum of \(-6.7\) and \(-7.6\) is \(-14.3\).

Step 1: Understand the Problem

We need to find the sum of \(-6.7\) and \(-7.6\). This involves adding two negative numbers.

Step 2: Combine Absolute Values

When adding negative numbers, we combine their absolute values. Here, the absolute values are \(|-6.7| = 6.7\) and \(|-7.6| = 7.6\). Add them together: \(6.7 + 7.6\).

Step 3: Add the Absolute Values

Calculate the sum of the absolute values: \(6.7 + 7.6 = 14.3\).

Step 4: Apply the Negative Sign

Since both original numbers were negative, the sum will also be negative. Therefore, the final answer is \(-14.3\).

Key Concepts

Understanding Absolute Value

Absolute value is a concept in mathematics that measures the distance of a number from zero on the number line, regardless of its direction. This means that the absolute value of both positive and negative numbers will always be a positive number. For example, the absolute value of \(-6.7\) is \(|-6.7| = 6.7\), and similarly, the absolute value of \(-7.6\) is \(|-7.6| = 7.6\).

Using absolute values is particularly helpful when adding negative numbers. Instead of focusing on their direction (negative), we first treat them like positive numbers to simplify the addition process. This involves working with their magnitudes alone, which makes calculations easier and avoids potential confusion with the signs.

Working with Negative Numbers

Negative numbers can sometimes seem tricky, but they are simply numbers located on the left side of zero on the number line. The main thing to remember is that negative numbers decrease in value as they move further from zero. Thus, \(-7.6\) is smaller than \(-6.7\).

When adding negative numbers, the rules differ slightly from adding positive numbers. Since both \(-6.7\) and \(-7.6\) are negative, their sum will also be negative. The process involves initially ignoring the negative signs, adding their absolute values, and then reapplying the negative sign to the result. This preserves the integrity of values and helps ensure that the additional negative nature of the numbers is accurately captured in the solution.

Basics of Integer Addition

Integer addition is one of the fundamental operations in arithmetic. It includes adding positive numbers, negative numbers, and zero. Here’s a quick refresher:

• Positive + Positive = Add the numbers normally.
• Negative + Negative = Add their absolute values, but keep the negative sign.
• Positive + Negative (or Negative + Positive) = Subtract the smaller absolute value from the larger, and take the sign of the number with the larger absolute value.

In our example, \(-6.7 + (-7.6)\), we are adding two negative numbers together. According to the rules, we add their absolute values: \(6.7 + 7.6 = 14.3\), and then apply the negative sign back to the result, resulting in an answer of \(-14.3\). This direct approach helps manage negatives more comfortably and offers consistently reliable results.