Problem 69 - Prealgebra and Introductory Algebra

Question

Use the given property of addition to complete the statement. The Inverse Property of Addition $$?+(-18)=0$$

Step-by-Step Solution

Answer

The number that satisfies the Inverse Property of Addition for -18 is 18.

Step 1: Understand the Inverse Property of Addition

The Inverse Property of Addition states that every real number has an additive inverse (a number that has the opposite sign). When a number and its additive inverse are added together, the result is zero. So, \(x + (-x) = 0\).

Step 2: Identify the additive inverse

In the statement \(? + (-18) = 0\), -18 is the given number and we are to find its additive inverse, which is the number that makes the result when added with -18 equals to zero. Let's call this unknown number 'a'.

Step 3: Find the additive inverse

From the additive inverse property of addition, we know that a number and its negative (or additive inverse) sum up to zero. Therefore, if -18 is our number, its additive inverse 'a' would satisfy the equation \(a + (-18) = 0\). Solving this equation for 'a' gives us \(a = 18\).

Key Concepts

Additive Inverse

The concept of an additive inverse is quite fundamental in algebra. It refers to a number that, when added to a given number, results in a sum of zero. For example, if you have the number -18, its additive inverse is 18. Why? Because when you add -18 and 18 together, you get 0, which is the identity element for addition.

To make it even simpler:

• The additive inverse of any positive number is its negative counterpart.
• Likewise, the additive inverse of a negative number is its positive counterpart.
• The additive inverse of zero is zero itself, as 0 + 0 equals 0.

Understanding the additive inverse helps in solving equations efficiently and is particularly useful when balancing equations in algebra.

Real Numbers

Real numbers encompass a broad spectrum of numbers that include all the rational numbers, such as integers and fractions, and all the irrational numbers which can't be expressed as a simple fraction. The subset of real numbers we typically deal with in elementary algebra includes:

• Whole numbers (e.g., 0, 1, 2, 3).
• Integers, both positive and negative (e.g., -3, -2, -1, 0, 1, 2).
• Fractions and decimals.
• Infinite non-repeating and non-terminating decimals (like \pi and the square root of 2).

When we talk about the inverse property of addition, it applies throughout the real numbers set. For any real number \(x\), there exists an inverse \(-x\) such that \(x + (-x) = 0\). This ensures that all fundamental operations in algebra hold true across the set of real numbers.

Addition Properties

Addition properties are basic principles in mathematics that define how exactly addition works. These properties are essential for understanding and solving algebraic problems, and they include:

• Commutative Property: States that the order in which you add two numbers doesn't matter. That is, \(a + b = b + a\).
• Associative Property: States that the way in which numbers are grouped when added doesn't change the sum. For instance, \((a + b) + c = a + (b + c)\).
• Identity Property: States that any number added to zero is the number itself. So, \(a + 0 = a\).
• Inverse Property: As discussed, it states every real number has an additive inverse which, when added, results in zero. That is, \(a + (-a) = 0\).

Recognizing and utilizing these properties can greatly simplify solving equations and understanding algebraic expressions.

Elementary Algebra

Elementary algebra is the branch of mathematics that deals with the basic structure and behavior of real numbers through the use of symbols and letters. This field introduces the concept of variables and algebraic expressions, allowing for the generalization and abstraction of arithmetic operations.In elementary algebra, we learn to:

• Utilize variables to represent numbers in expressions and equations. For example, in the equation \(? + (-18) = 0\), the symbol '?' is a placeholder for the unknown value.
• Apply algebraic properties, such as the inverse property of addition, to manipulate expressions and solve equations.
• Understand and construct equations, find their solutions, and graph relationships on the coordinate plane.

Elementary algebra equips students with the toolkit needed to solve a variety of mathematical problems and lays the groundwork for more advanced studies in mathematics.