Problem 80 - Algebra A Combined Function

Question

Perform the indicated operation. 6+(-15)

Step-by-Step Solution

Answer

The result of 6 + (-15) is -9.

Step 1: Identify the operation

The problem given is an addition of two numbers: 6 and -15. The operation to be performed here is the addition of a positive number and a negative number.

Step 2: Understand signed numbers

When adding a positive number and a negative number, think of it as finding the difference between the two numbers. Since 15 is greater than 6 and it carries a negative sign, the result will be negative.

Step 3: Subtract the numbers

Subtract the smaller absolute value from the larger absolute value. Subtract 6 from 15: \(15 - 6 = 9\).

Step 4: Determine the sign of the result

Since the number with the larger absolute value (15) is negative, the result of our operation will also be negative. Thus, the result of adding 6 and -15 is -9.

Key Concepts

Addition of Positive and Negative Numbers

Addition of signed numbers, particularly when mixing positive and negative digits, can initially seem confusing, but it's intuitive once you get the hang of it. Here's the trick: when adding a positive number to a negative number, it's like finding the difference between them.

• Visualize a number line. Positive numbers move you to the right, while negative numbers move you to the left.
• When you have both, determine which number has the larger absolute value. This tells you the net direction you'll move.
• If the larger absolute value is negative, the result of the addition will be negative.

The exercise asks us to add 6 and -15. Here, 15 has a greater absolute value than 6, meaning our result will lean in the negative direction. Thus, the result of 6 + (-15) is -9.

Absolute Value

Understanding absolute value is pivotal when dealing with operations involving signed numbers. The absolute value of a number is its magnitude, regardless of sign.

• Think of absolute value as the distance from zero on a number line.
• It is always a positive number or zero, because distance can’t be negative.
• For example, both 15 and -15 have an absolute value of 15.

Using absolute values helps us decide which number is larger in magnitude when combining positive and negative numbers. In our exercise, 15 and -15 have the same absolute value, but their relevance in the problem is that -15 is the larger negative going in the opposite direction than 6. This guides our final sign and value adjustment.

Subtracting Numbers

Subtracting numbers, especially when involving signed values, is closely related to addition operations. When we think of subtracting, we often imagine removing or decreasing value. However, with signed numbers, subtraction can simply mean adjusting direction on the number line.

• To subtract a number, we add its negative equivalent. For example, subtracting −5 is like adding 5.
• In our problem, we simplify 6 + (-15) by subtracting, because we consider the difference between the absolute values. Thus, 15 minus 6 equals 9.
• Determining which absolute value is greater helps us decide the sign of the final result, leading to a negative outcome in this scenario.

This method allows us to convert complex additions into simpler subtractions, making the calculation process more straightforward.