Problem 5 - Algebra for College Students

Question

Perform the following operations with real numbers. $$-8-14$$

Step-by-Step Solution

Answer

The result is -22.

Step 1: Understanding the Problem

The problem asks us to perform the operation of subtracting 14 from -8. This involves dealing with negative numbers and understanding how subtraction works with them.

Step 2: Identify the Numbers

Identify the numbers involved in the operation. We have two numbers: -8 (the minuend) and 14 (the subtrahend), and we need to simplify the expression \(-8 - 14\).

Step 3: Change Subtraction to Addition

When dealing with subtraction, it can be helpful to transform the operation into an addition for simplification. Subtracting a positive number is equivalent to adding its negative. Therefore, \(-8 - 14\) can be rewritten as \(-8 + (-14)\).

Step 4: Perform the Addition

Now, add the two negative numbers together. When adding negative numbers, we simply add their absolute values and keep the negative sign. So, \(-8 + (-14) = -22\).

Step 5: Write the Final Answer

After calculating, we find that the result of \(-8 - 14\) is \(-22\). Thus, the simplified result is \(-22\).

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero. They are typically represented with a minus sign (-) in front of them. In many real-world scenarios, negative numbers are encountered, such as in temperatures below zero or financial debts.
When handling negative numbers, there are a few important rules to remember:

• Adding a negative number is the same as subtraction.
• Subtracting a negative number means you are essentially adding a positive number.
• Multiplying or dividing two negative numbers results in a positive number.
• Multiplying or dividing a positive and a negative number results in a negative number.

These rules are crucial for effectively solving mathematical problems involving negative numbers.

Subtraction

Subtraction is the process of taking one number away from another. It is often represented by the minus sign (-). In the context of the exercise we have, subtraction is taking away 14 from -8.
The key points to consider when subtracting are:

• Identify the minuend (the number from which another number is subtracted) and the subtrahend (the number to be subtracted).
• The operation can often be simplified by changing subtraction to addition, especially when negative numbers are involved. This makes calculations easier and reduces errors.
• For instance, \(-8 - 14\) can be transformed to \(-8 + (-14)\).

Understanding this transformation is important, as it allows us to handle both positive and negative numbers more intuitively.

Addition

Addition involves combining numbers to get a sum. When dealing with negative numbers, addition may require special attention, especially when both numbers to be added are negative.
For example, in the expression \(-8 + (-14)\), both numbers are negative. The rule for adding negative numbers is straightforward:

• Find the absolute values of each number (ignore the negative sign and see them as positive values).
• Add these absolute values.
• Then, place a negative sign in front of the sum.

This will result in \(-8 + (-14) = -22\). Remember, the sum of two negative numbers is always negative and its absolute value is the sum of their absolute values.