Problem 8 - Algebra for College Students

Question

Perform the following operations with real numbers. $$8-22$$

Step-by-Step Solution

Answer

The result of \(8 - 22\) is \(-14\).

Step 1: Identify the Operation

The problem requires the subtraction of two real numbers, which is represented by the expression \(8 - 22\).

Step 2: Subtract the Numbers

When subtracting, adjust the smaller number from the larger one. Here's what that looks like for our numbers: \(8 - 22\) can be rewritten by changing subtraction into adding a negative number, so it's equivalent to \(8 + (-22)\).

Step 3: Simplify the Expression

Add \(8 + (-22)\). Since 22 is larger and negative, subtract 8 from 22 and apply the negative sign. Calculate: \(22 - 8 = 14\), and apply the negative sign, getting \(-14\). The result is: \(8 - 22 = -14\).

Key Concepts

Subtraction

Subtraction is one of the four basic arithmetic operations and is essentially the process of removing or "taking away" one quantity from another. In general terms, when you subtract, you are looking at the difference between two numbers. To understand subtraction with real numbers, it's helpful to keep a few core ideas in mind:

• The order matters: Subtracting 5 from 10 is different than subtracting 10 from 5.
• We often describe subtraction using the terms "minuend" (the starting number) and "subtrahend" (the amount to subtract).
• Subtraction can be visualized on a number line by counting backward from the minuend by the subtrahend.

In our example, we have the expression \(8 - 22\). Here, 8 is the minuend and 22 is the subtrahend. We can think of subtraction as adding the opposite, which brings us to the concept of negative numbers.

Negative Numbers

Negative numbers are numbers less than zero, and they can sometimes be a bit tricky to grasp at first, particularly in the context of subtraction. These numbers are essential in representation of debts, losses, or temperatures below freezing. When working with negative numbers, here are some points to consider:

• A negative number has a minus sign (-) in front, such as \(-2\) or \(-10\).
• When you subtract a number, it is the same as adding its opposite. Hence, subtracting a positive number results in a decrease, while subtracting a negative number results in an increase.
• Adding \(-22\) to 8, as shown in our step-by-step solution, involves combining positive and negative values. This requires us to consider the absolute values to perform a proper operation.

In the case of our example, we consider \(-22\) as the opposite of 22, leading us to simplify the subtraction to evaluate \(8 + (-22)\). Handling negative numbers correctly is the key to accurately solving expressions involving subtraction.

Expression Simplification

Expression simplification is the process of reducing a mathematical expression into its simplest form. This involves combining like terms, performing operations, and sometimes rearranging components for clarity. Here are a few crucial points:

• To simplify expressions, it's important to follow the order of operations (PEMDAS/BODMAS).
• When simplifying, ensure that all signs (positive or negative) are carried through accurately.
• Pay close attention to negative signs during simplification, as they often change the outcome of the calculations.

In our solved exercise, the expression \(8 + (-22)\) is simplified by considering the larger number’s absolute value, which is 22, subtracting the smaller number’s absolute value, which is 8, and then applying the negative sign of the larger absolute value result (since 22 is negative, the result is negative). Therefore, the expression simplifies to \(-14\). Simplifying expressions correctly ensures that our final answer is both understandable and accurate.