Problem 46 - Introductory and Intermediate Algebra for College Students 4th

Question

Find each sum without the use of a number line. $$6.4+(-35)+(-2.6)+14$$

Step-by-Step Solution

Answer

-17.2

Step 1: Separate Positive and Negative Numbers

First step is identifying and separating the positive and negative numbers from the equation. So, we have \(6.4\) and \(14\) as our positive numbers and \(-35\) and \(-2.6\) as our negative numbers.

Step 2: Sum of Positive and Negative Numbers

Now, we sum the positive numbers together and the negative numbers together separately. This means we do the following calculations: \(6.4 + 14 = 20.4\) (sum of positive numbers) and \(-35 -2.6 = -37.6\) (sum of negative numbers).

Step 3: Final Summation

Finally, we add the sum of the positive numbers to the sum of the negative numbers: \(20.4 + (-37.6) = -17.2\)

Key Concepts

Addition of Real Numbers

Addition of real numbers is an essential concept in mathematics. Real numbers include both rational and irrational numbers, ranging from integers like -2, 0, and 3, to decimals like 6.4 and \(-2.6\). When adding real numbers, it's important to understand the operation's rules so you can handle both whole numbers and decimals. This involves combining numbers accurately while respecting their signs and unique values.

• If the numbers have the same sign (either both positive or both negative), add their absolute values together and keep the sign.
• If the numbers have different signs, subtract the smaller absolute value from the larger one and take the sign of the number with the larger absolute value.

Breaking down operations step by step ensures clarity and accuracy in your calculations.

Positive and Negative Numbers

Understanding positive and negative numbers is crucial when working with real numbers. Positive numbers are greater than zero and are typically written without a sign, like 6.4 and 14. Negative numbers, on the other hand, are less than zero and are expressed with a minus sign in front of them, such as \(-35\) and \(-2.6\).

Think of negative numbers as opposite to their positive counterparts on a number line, even when you're not using one visually.

• Positive numbers increase the value when added to another number.
• Negative numbers decrease the value when added to another number.

Hence, the sum of positive and negative numbers depends on their magnitudes: if positives have a greater magnitude, you end up with a positive sum, and vice versa.

Sum Calculation Methods

There are strategic methods to calculate the sum of a combination of positive and negative numbers. Summing involves a basic yet strategic process that begins by understanding and organizing your numbers correctly.

• First, separate positive numbers from negative numbers to manage them more easily.
• Next, perform separate summations for each group. For positive numbers, \( 6.4 + 14 = 20.4 \), and for negative ones \( -35 + (-2.6) = -37.6 \).

After computing the individual sums, calculate the final result by combining these subtotals: add the positive total to the negative (or vice versa), remembering to apply the rules for combining numbers of different signs.
In our exercise, this results in \( 20.4 + (-37.6) = -17.2 \), demonstrating how thoughtful separation and calculation can streamline solving even seemingly complex addition problems.