Problem 3 - Introductory and Intermediate Algebra for College Students 4th

Question

Consider the subtraction \(5-(-7)\) a. Find the opposite, or additive inverse, of \(-7\) b. Rewrite the subtraction as the addition of the opposite of \(-7\)

Step-by-Step Solution

Answer

a. The opposite, or additive inverse, of \(-7\) is \(+7\). b. The subtraction \(5 - (-7)\) can be rewritten as the addition of the opposite of \(-7\), or \(5 + 7\).

Step 1: Finding the Additive Inverse

The additive inverse, or opposite, of a number is the value that, when added to the original number, results in zero. Therefore, the opposite of \(-7\) is \(+7\), because \(-7 + 7 = 0\).

Step 2: Rewriting the Subtraction

The operation \(5 - (-7)\) can be rewritten as an addition, by substituting the subtraction of the number with the addition of its opposite. Thus, \(5 - (-7)\) equals to \(5 + 7\).

Key Concepts

Understanding the Subtraction of Negative Numbers

When dealing with subtraction of negative numbers, it can initially seem a bit counterintuitive. However, a helpful way to approach this is to consider the concept of additive inverses. If you have the subtraction problem like in the exercise, where you need to calculate \(5-(-7)\), you must first identify the additive inverse of \(-7\), which is \(+7\).

This is because an additive inverse is simply a number that, when added to the original number, equals zero. So, the equation \(-7 + 7 = 0\) confirms that \(+7\) is indeed the inverse of \(-7\). From here, you can transform the problem into one of addition rather than subtraction. This makes it clear that subtracting negative seven is the same as adding positive seven, simplifying to \(5 + 7 = 12\).

To further assist students, remember to emphasize that each positive number has a negative counterpart and vice versa. Thus, whenever they encounter a subtraction of a negative number, they can confidently convert it into addition.

The Basics of Addition

Moving on to the next crucial arithmetic operation, addition is one of the first operations we learn in math. Addition combines two or more numbers into a single sum. For example, adding \(5\) and \(7\), as seen in the modified equation from the exercise, results in \(12\). This principle doesn't change when considering the previously discussed rule about additive inverses and the subtraction of negative numbers.

It's also vital for students to appreciate that addition is commutative. This means the order in which numbers are added does not affect the sum. Hence, \(5 + 7\) is equal to \(7 + 5\). Encourage students to practice addition with different types of numbers—positive, negative, and even decimals—to gain fluency and confidence in solving equations across various mathematical funds.

Algebraic Expressions and Their Components

Lastly, let's discuss algebraic expressions, which are combinations of numbers, variables (letters that represent numbers), and arithmetic operations like addition, subtraction, multiplication, and division. For example, \(5 - (-7)\) is a basic algebraic expression before we apply the concept of additive inverse.

Understanding the structure of algebraic expressions is key to manipulating and simplifying them. In this case, recognizing that the subtraction of a negative number can be converted to addition is part of that simplification process. Breaking down algebraic expressions into their simplest forms allows for easier solving of math problems and leads to a better grasp of more complex algebraic equations. Always encourage exploration of these concepts with various terms and operations to build a solid foundation in algebra.