Problem 82 - Prealgebra

Question

The problems below review addition, subtraction, multiplication, and division of positive and negative numbers, as covered in this chapter. Perform the indicated operations. $$-8+4$$

Step-by-Step Solution

Answer

The result is -4.

Step 1: Understand the operation

We are asked to perform the addition of -8 + 4. When dealing with the addition of a negative number and a positive number, it is equivalent to subtracting the absolute values of the numbers: |-8| = 8 and |4| = 4, and then taking the sign of the larger absolute value.

Step 2: Determine the operation type

Since -8 is a negative number and 4 is a positive number, the operation is essentially -8 + 4, which means subtracting 4 from 8.

Step 3: Subtract the smaller number from the larger number

Subtract the smaller absolute value (4) from the larger absolute value (8): 8 - 4 = 4.

Step 4: Determine the sign

The number with the larger absolute value is -8. Since 8 is larger than 4, and the larger number's original sign was negative (-8), the result takes on this sign, making the operation result in -4.

Key Concepts

Addition of Integers

Adding integers might initially seem daunting, especially when dealing with both positive and negative numbers. But understanding this concept can be simplified by considering absolute values and signs.

• Combining Signs: When you add two numbers with the same sign, whether both positive or negative, simply add their absolute values and keep the common sign. For example, adding -5 and -3 gives you -8 because you add 5 and 3, then apply the negative sign.
• Different Signs: When the signs differ, like in -8 and 4, treat it as a subtraction. Subtract the smaller absolute value from the larger one and use the sign of the number with the larger absolute value. Here, between -8 and 4, the absolute value of 8 is greater, so the result will be negative.

Trying to see the operation as a tug-of-war can be helpful. Think of negatives as pulling in one direction, and positives in the other. The side with greater force determines the sign of the sum.

Subtraction with Negative Numbers

Subtraction of integers, especially with negatives involved, can be simplified by rewriting it as addition.

• Transform Subtraction: Change any subtraction sign into an addition one, and alter the sign of the number following it. For instance, -8 + 4 can be seen as adding -8 and 4 since -8 already includes the negative sign.
• Addition Rule: Perform the operation by taking note of signs. If the operation changes to adding a negative, like in some cases of subtraction, simply follow the rules of addition.

By transforming subtraction problems into addition, you can apply consistent rules across both types of operations, yielding more straightforward computations.

Absolute Value Comparison

Absolute value plays a crucial role in simplifying integer operations. It represents the magnitude of a number without considering its sign.

• Definition: Absolute value is the number's distance from zero on a number line, always taking a positive form. For example, both |-8| and |8| equal to 8.
• Comparison: During operations like addition, it helps to determine which number is larger. The absolute value comparison tells us which integer will dominate the operation's result.

Understanding and using absolute values allow for better handling of when and how signs are applied to result from integer operations, leading to fewer mistakes and clearer results.