Problem 56 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each series of additions and subtractions. $$-9-5+4-17$$

Step-by-Step Solution

Answer

-27

Step 1: Apply the Rule of Signs for Addition and Subtraction

For addition and subtraction, the rule of signs is: 'Adding a positive number is same as moving to the right on the number line. Adding a negative number is the same as moving to the left on the number line. Subtracting a positive number is the same as adding a negative number; subtracting a negative number is the same as adding a positive number.' Calculating the provided numbers step by step is therefore equivalent to starting at -9 on the number line, then moving to the left five spaces (adding -5), right 4 spaces (adding 4), and finally left 17 spaces (adding -17).

Step 2: Combine Numbers

When we combine these movements, we find that: \( -9 + (-5) + 4 + (-17) = -27 .\)

Key Concepts

Rule of Signs

Understanding the rule of signs is crucial for simplifying algebraic expressions. Basically, it tells us how the signs of numbers change when we perform addition or subtraction. When adding two numbers:

• If their signs are the same, we add their absolute values and keep the sign.
• If their signs are different, we subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value.

For subtraction:

• Subtracting a positive number is equivalent to adding its negative counterpart.
• Similarly, subtracting a negative number is the same as adding a positive number of the same absolute value.

This rule helps us to understand and perform operations without errors, especially when dealing with multiple positive and negative numbers in an expression.

Number Line Operations

Using a number line can be a great way to visualize the process of adding and subtracting numbers, especially when dealing with positive and negative values. Imagine a horizontal line where each position represents a number, with zero in the middle. As you move to the right, the numbers increase (positive direction), and as you move to the left, the numbers decrease (negative direction).

• To add a positive number, move to the right from your starting point.
• To add a negative number, move to the left.
• Subtraction is the reverse: to subtract a positive number, move to the left, and to subtract a negative number, you move to the right.

Applying this to our exercise, we start at -9 and visually track our movements left and right to end up at our final result.

Combining Like Terms

Combining like terms is a method used to simplify algebraic expressions by merging terms that have the same variable raised to the same power. However, even without variables, this concept still applies when working with numbers alone, as in our exercise.

• Add up all the positive values.
• Add up all the negative values separately.
• Combine these two totals to find the final result.

In the exercise's context, we treat each number as a term, combine all negative 'terms' together, and then the remaining positive 'term' to simplify the expression to a single number.

Algebraic Addition and Subtraction

Algebraic addition and subtraction pertain to operations with algebraic terms. In practice, it involves:

• Identifying like terms.
• Applying the rule of signs to determine how terms combine.
• Making sure that variables and their exponents match when combining terms.

Our exercise, although it does not involve variables, still shows the underlying principle: numbers can be combined following specific rules to simplify the expression. We can think of each number as a term, where like terms are simply all the positive numbers and all the negative numbers, and we combine them accordingly to reach a simplified result.