Problem 5 - Intermediate Algebra

Question

Perform the following operations with real numbers. $$ 8+(-15) $$

Step-by-Step Solution

Answer

The result of \(8 + (-15)\) is \(-7\).

Step 1: Identify the numbers and signs

The given expression is \(8 + (-15)\). Here, \(8\) is a positive number and \(-15\) is a negative number. We need to perform an addition operation between them.

Step 2: Rewrite the expression using subtraction

Since adding a negative number is the same as subtracting its positive counterpart, the expression \(8 + (-15)\) can be rewritten as \(8 - 15\).

Step 3: Perform the subtraction

Subtract \(15\) from \(8\). Since \(8\) is less than \(15\), the result will be negative. Calculate \(15 - 8 = 7\), hence the answer will be \(-7\).

Key Concepts

Addition of Negative Numbers

When dealing with the addition of negative numbers, the process can sometimes feel a bit tricky. But don't worry! It's simpler than it seems. Let's break it down into easy steps:

• When you see an expression like \(8 + (-15)\), it involves adding a positive number to a negative number.
• The key here is to understand that adding a negative number is the same as subtracting the positive version of that number.

Therefore, the expression \(8 + (-15)\) can be reinterpreted as \(8 - 15\). Think of the negative sign as a signal to switch from addition to subtraction. This small mental adjustment makes calculations with negative numbers more intuitive.

Subtraction Operation

Subtraction is one of the fundamental operations in mathematics, and it's integral to understanding how real numbers interact with each other. Let's take a closer look at this operation:

• Subtraction involves taking away a certain quantity from another. In mathematical terms, it means finding the difference between two numbers.
• In the example expression \(8 - 15\), we are subtracting a larger number (15) from a smaller number (8).

This situation demonstrates a subtraction operation resulting in a negative outcome because the subtrahend (15) is greater than the minuend (8). As a rule of thumb, when a smaller number is subtracted from a larger number, the result is positive, but when it's the other way around, as in this case, the result is negative.

Integer Operations

Integer operations involve the arithmetic manipulation of whole numbers, both positive and negative. Understanding integer operations is crucial for mastering problems involving real numbers.

• Integers include both positive numbers, negative numbers, and zero.
• Basic operations, such as addition, subtraction, multiplication, and division, all apply to integers, each with specific rules.

When performing these operations, especially addition and subtraction, use the following guidelines:

• If both numbers have the same sign, add their absolute values and keep the common sign.
• If the numbers have different signs, subtract the smaller absolute value from the larger one, and use the sign of the number with the larger absolute value for the result.

These operations are the backbone of more complex mathematical concepts, so it's important to grasp these fundamentals well.