Problem 7 - Elementary and Intermediate Algebra

Question

For each addition, just determine the sign of the answer. a. \(39.6+(-64.9)\) b. \(-18.9+19.8\)

Step-by-Step Solution

Answer

a. Negative b. Positive

Step 1: Understanding Positives and Negatives

When adding a positive number to a negative number, you essentially subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value.

Step 2: Analyzing Part a

For part (a), we have the numbers 39.6 and -64.9. The absolute value of 64.9 is greater than 39.6. Therefore, the result will have the sign of -64.9, which is negative.

Step 3: Analyzing Part b

For part (b), we have the numbers -18.9 and 19.8. The absolute value of 19.8 is greater than 18.9. Therefore, the result will have the sign of 19.8, which is positive.

Key Concepts

Positive and Negative Numbers

In the realm of integers, understanding positive and negative numbers is pivotal. Positive numbers are those that are greater than zero. For example, numbers like 2, 15, or 238 are all positive. These numbers are typically straightforward and are added just as we do in regular arithmetic.
Negative numbers, on the other hand, are numbers that are less than zero. You can recognize them by the minus (-) sign that precedes them, such as -3, -100, or -0.5. These numbers convey a quantity less than nothing, which can be seen in contexts such as temperatures below freezing.

• Positive: Sign is '+' or no sign displayed (e.g., 7, 12.5)
• Negative: Sign is '-' indicating below zero (e.g., -4, -22.1)

When you add a negative number, it's like subtracting its positive equivalent. It's crucial to know the nature of the numbers to correctly predict the summation outcomes.

Absolute Value

Absolute value is an important concept that helps in determining the magnitude of a number, without regard to its sign. Represented as \(|x|\), where \(|x|\) means the absolute value of x, it describes the distance a number is from zero on the number line, irrespective of direction.
For instance:

• The absolute value of 3 is 3, represented as \( \|3\| = 3 \)
• For a negative number like -8, the absolute value is 8: \( \|-8\| = 8 \)

This concept becomes particularly useful in operations involving both positive and negative numbers. Absolute values help in determining which number has a larger or smaller magnitude, disregarding whether the number is positive or negative, making it easy to calculate final outcomes.

Sign Determination in Addition

When performing addition with both positive and negative numbers, determining the sign of the result is essential yet simple. The key is to compare the absolute values of the numbers involved. If the larger absolute value belongs to a positive number, the result is positive. Conversely, if the larger absolute value belongs to a negative number, the result is negative.

Here's a quick guide:

• Add a positive and negative number: Subtract the absolute value of one from the other.
• If a negative number has the larger absolute value, the sum is negative.
• If a positive number has the larger absolute value, the sum is positive.

By closely observing the examples provided, such as \(39.6+(-64.9) \) and \(-18.9+19.8 \), this method helps in easily determining the sign of the result by comparing the absolute values. This approach simplifies processing the addition of mixed sign numbers effectively.