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

Question

Consider the subtraction \(2-(-8)\) a. Find the opposite, or additive inverse, of \(-8\) b. Rewrite the subtraction as the addition of the opposite of \(-8\)

Step-by-Step Solution

Answer

a. The additive inverse of \(-8\) is \(8\). b. The subtraction \(2-(-8)\) can be rewritten as \(2+8\).

Step 1: Find the additive inverse of -8

To find the additive inverse of a number, change its sign. So, the additive inverse of \(-8\) is \(8\).

Step 2: Rewrite the original subtraction

We are asked to rewrite the subtraction \(2-(-8)\) as an addition. We use the result from step 1, so we have \(2+8\).

Key Concepts

Additive Inverse

The concept of an additive inverse helps us understand how to turn subtraction into addition, which can simplify many math operations. Simply put, the additive inverse of a number is another number that, when added to the original number, results in zero. This operation changes the sign of the number. For instance, the additive inverse of

• -8 is 8
• 8 is -8

This transformation is essential in algebra when dealing with subtraction of negative numbers. By converting a subtraction problem into an addition problem, math becomes more straightforward. This is because addition has properties like being associative and commutative, which aren’t as easy to apply directly to subtraction operations.

Integer Operations

Integer operations cover addition, subtraction, multiplication, and division involving whole numbers. These operations demand a solid grasp of the rules governing positive and negative numbers, which often stumble learners. When working with integers:

• Adding two positive numbers results in a positive number.
• Adding two negative numbers results in a negative number.
• Subtracting a negative number is the same as adding its additive inverse.
• Subtracting a positive number decreases the value.
• Multiplying or dividing two numbers with the same sign yields a positive result.
• Multiplying or dividing two numbers with different signs yields a negative result.

In the specific problem of subtracting \(2-(-8)\), you switch the operation to addition by using the additive inverse, resulting in the operation \(2 + 8\), which equals 10. Mastery of these rules simplifies problem-solving in algebra and beyond.

Algebraic Expressions

Algebraic expressions incorporate numbers, variables, and operations, offering a unifying language for mathematics. Understanding how to manipulate these expressions is foundational for algebraic problem-solving. They often include:

• Variables, which are symbols used to represent unknown or arbitrary numbers.
• Constants, which are fixed numbers.
• Operations such as addition, subtraction, multiplication, and division.

In the context of subtraction, transforming an algebraic expression through the use of additive inverses can simplify solving equations. For example, if you have the expression like \( x - (-y) \), it can be rewritten using the additive inverse of \(-y\) to \( x + y \). This simplification can make the problem more manageable and aligns with well-understood arithmetic rules. Grasping how to systematically reconfigure such expressions is crucial for tackling equations and understanding algebraic functions.