Problem 17 - Prealgebra

Question

Combine the following by using the rule for addition of positive and negative numbers. $$5+(-8)$$

Step-by-Step Solution

Answer

The result is -3.

Step 1: Identify the Numbers

We have two numbers to combine: 5 (a positive number) and -8 (a negative number).

Step 2: Apply the Rule for Addition of Positive and Negative Numbers

When adding a positive number and a negative number, you subtract the smaller number from the larger absolute value number and keep the sign of the number with the larger absolute value. Here, compare the absolute values: |5| = 5 and |-8| = 8. Since 8 is larger, the result will take the sign of -8.

Step 3: Calculate the Result

Subtract the smaller absolute value (5) from the larger absolute value (8) which gives us 3. Keep the sign of the number with the larger absolute value, which is negative. Thus, the result is -3.

Key Concepts

Understanding Positive and Negative Numbers

In mathematics, numbers are classified into two main types: positive and negative numbers. Positive numbers are greater than zero and are typically written without any sign, for example, 1, 2, and 3. Negative numbers are less than zero and are represented with a minus sign, such as -1, -2, and -3.

When you visualize numbers on a number line, positive numbers are found to the right of zero, moving further into larger values. Negative numbers, on the other hand, are to the left of zero, getting smaller as they move leftward. This positioning is crucial for understanding how operations like addition and subtraction work.

• Positive numbers increase value when added.
• Negative numbers decrease value when added.

Accounting for both types of numbers is essential in integer operations, especially when learning to correctly apply rules for the addition of integers.

The Importance of Absolute Value

Absolute value plays an important role in integer operations, especially when dealing with positive and negative numbers. The absolute value of a number is its distance from zero on the number line, regardless of direction. This means it is always a non-negative number. For instance, the absolute value of both 5 and -5 is 5.

Understanding absolute value helps us compare the size of different numbers without considering their sign. In the context of adding integers like 5 and -8, we first look at their absolute values: |5| = 5 and |-8| = 8.

• If two numbers have the same absolute value, they are equidistant from zero.
• The number with the larger absolute value determines the sign of the result in operations that involve both positive and negative numbers.

Utilizing absolute value makes it easier to grasp how operations will affect numbers on the number line.

Mastering Integer Operations

Integer operations, such as addition and subtraction, often require an understanding of both the signs and absolute values of numbers involved. Adding integers with different signs can be tricky, but a simple rule can help.

When you add a positive and a negative number, like 5 and -8, you compare their absolute values to determine which has the greater magnitude. In our example:

• The absolute value of -8 (8) is larger than the absolute value of 5 (5).
• Subtract the smaller absolute value from the larger: 8 - 5 = 3.
• Since the larger absolute value corresponds to -8, the result is negative, making the answer -3.

Practicing this process with different numbers will strengthen your ability to confidently handle integer operations, ensuring each result accurately reflects the relationships on the number line.