Problem 26 - Elementary and Intermediate Algebra

Question

Add. Do not use the number line except as a check. \(8+(-5)\)

Step-by-Step Solution

Answer

3

Step 1: Identify the numbers

Identify the two numbers that need to be added: 8 and -5.

Step 2: Recognize the operation

Recognize that this is a case of adding a positive number (8) and a negative number (-5).

Step 3: Determine the net effect

Since one of the numbers is negative, subtract the smaller number from the larger number and keep the sign of the larger number. Here, subtract 5 from 8, which results in 3.

Step 4: Assign the correct sign

Based on the rule, since 8 is larger and it is positive, the final result is positive. Therefore, the answer is 3.

Key Concepts

Adding Positive and Negative Numbers

When we add a positive number to a negative number, we are essentially combining values with different signs. Imagine the positive number as taking steps forward and the negative number as taking steps backward. The result depends on which number is larger in magnitude and their respective signs. In the example of adding 8 and -5, we subtract the smaller magnitude from the larger one, thus: 8 - 5. Since 8 is positive and larger than 5, the result is a positive 3.

Remember these rules for adding positive and negative numbers:

• If the signs are different, subtract the smaller magnitude from the larger and keep the sign of the larger number.
• If the signs are the same, simply add the magnitudes and keep the common sign.

Integer Operations

Integers include all whole numbers and their negatives, like -2, -1, 0, 1, 2. Integer operations involve addition, subtraction, multiplication, and division among these numbers. For addition, particularly with different signs, the rule is to combine their magnitudes while keeping track of their signs.

For example, adding 8 and -5 involves these steps:

• Identify the numbers: 8 and -5.
• Recognize you're adding a positive and negative number.
• Subtract 5 from 8: 8 - 5 = 3.
• Since 8 is larger and positive, the result is positive 3.

Practicing these steps helps in mastering how to handle other integer operations too.

Basic Arithmetic

Basic arithmetic is the foundation of all math. It includes addition, subtraction, multiplication, and division. Understanding how to add positive and negative numbers is crucial for solving more complex problems.

In our example:

• First, identify the values and their signs: 8 (positive), -5 (negative).
• Next, combine them accordingly: we subtract the smaller magnitude from the larger one.
• Lastly, apply the sign of the larger number: positive 8.

Learning these fundamental steps ensures that we can approach more complicated arithmetic with confidence.

Subtraction in Addition

Subtraction often appears when adding integers with different signs. Essentially, adding a negative number is like subtracting its positive counterpart. For instance, 8 + (-5) can be thought of as 8 - 5.

Steps for using subtraction in addition:

• Identify each number's value and sign: 8, -5.
• Subtract the smaller absolute value from the larger one: 8 - 5.
• Determine the sign of the result based on the larger number: result is 3 and positive.

This method is invaluable for dealing with varied arithmetic problems, providing clear and consistent results.