Problem 1 - Algebra for College Students

Question

Perform the following operations with real numbers. $$8+(-15)$$

Step-by-Step Solution

Answer

The result is -7.

Step 1: Identifying the Operation

The given operation is between two numbers, 8 and -15. We need to add these two numbers. Adding a negative number is the same as subtracting its positive counterpart.

Step 2: Applying the Rule of Addition with Negative Numbers

When adding a positive number and a negative number, subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value. Here, the absolute values are 8 and 15. So, we subtract the smaller from the larger: 15 - 8 = 7.

Step 3: Determining the Sign

Since the negative number (-15) has a larger absolute value than the positive number (8), the result takes the sign of the negative number. Therefore, the result of the operation is -7.

Key Concepts

Addition of Real Numbers

When we talk about adding real numbers, it's important to know that these numbers include both positive and negative numbers, as well as fractions, decimals, and whole numbers. Addition with real numbers follows a straightforward approach but can involve different scenarios:

• Adding two positive numbers always results in a positive number.
• Adding two negative numbers results in a negative number. Here, you add their absolute values and put a negative sign in front of the result.
• Adding a positive number to a negative number requires some extra care, which we'll explore more below.

In our exercise, we see the operation 8 + (-15). Here, we combine a positive and a negative number, which means we have to consider the absolute values and the signs carefully. It's critical to focus on the values explicitly before worrying about positive or negative signs, ensuring accurate solutions.

Negative Numbers

Negative numbers represent values less than zero and are crucial in understanding real number operations. They have unique properties as compared to positive numbers:

• Negative numbers are found on the left side of zero on a number line. As you move left, their value actually decreases.
• When adding a negative number to a positive number, you are effectively subtracting the absolute value of the negative number from the positive number.
• Negative numbers have higher absolute values, but their effectiveness or impact is weaker; for example, comparing -2 and -5, -2 has a larger value.

In mathematics, understanding these distinctions helps. As per the exercise, knowing that 8 + (-15) equates to subtracting 15 from 8 simplifies the process and shows the importance of sign management. Implementing these properties allows easier calculations in complex problems.

Absolute Value

Absolute value is a concept that tells us how far a number is from zero on a number line, ignoring direction. This means:

• The absolute value of any positive number is the number itself.
• The absolute value of a negative number is the number without its negative sign.
• Essentially, absolute value turns every number into a non-negative version of itself.

Absolute value is symbolized by vertical bars around a number, like \(|x|\). In our exercise of adding 8 + (-15), we look at |8| which is 8 and |-15| which is 15. This understanding is crucial as it determines how you manipulate numbers in operations like addition and subtraction. The rule of thumb is to subtract the smaller absolute value from the larger when combining numbers with different signs, then assign the sign of the number with the larger absolute value. In our problem, knowing that the absolute value of -15 is greater influenced the final negative sign of our result, which is -7.