Problem 68 - Elementary and Intermediate Algebra

Question

Add. $$ -25+25 $$

Step-by-Step Solution

Answer

0

Step 1: Identify the Numbers

Examine the numbers in the expression to determine what needs to be added. Here, we have two numbers: -25 and +25.

Step 2: Understand the Concept of Additive Inverses

Realize that +25 is the additive inverse of -25. Additive inverses are two numbers whose sum is zero. When you add -25 and +25, they will cancel each other out.

Step 3: Perform the Addition

Add -25 to +25: \[-25 + 25 = 0\]Since +25 is the opposite of -25, their sum is 0.

Key Concepts

Integer Addition

Integer addition involves combining whole numbers, which can be positive, negative, or zero. When adding integers, it's important to identify the sign associated with each number. The sign indicates whether the number contributes positively or negatively to the sum. For example:

• Positive numbers, like \(+5\), add value.
• Negative numbers, like \(-5\), subtract value.

When adding numbers with different signs, their absolute values (the number without a sign) are compared, and the smaller is subtracted from the larger. The result takes the sign of the number with the larger absolute value. However, when numbers are exactly opposite, like \(-25\) and \(+25\), they sum to zero.

Zero Property

The zero property in mathematics refers to the idea that any number added to zero remains unchanged. More specifically in the context of additive inverses, it means that any number and its opposite (additive inverse) sum up to zero. This key concept is vital in understanding why certain arithmetic operations yield zero. For example:

• The equation \(-25 + 25 = 0\) utilizes the zero property, because the numbers are inverses.
• The same occurs with smaller numbers like \(-1 + 1 = 0\).

The zero property simplifies many algebraic equations and is an important foundation in algebra basics, ensuring students understand how and why numbers cancel each other out.

Algebra Basics

Algebra basics involve understanding how numbers and symbols interact. At its core, algebra replaces numbers with letters or symbols to solve problems. When dealing with integer addition and the zero property, understanding algebra basics helps explain how equations function. Key points include:

• Using variables, such as \(x\) or \(y\), to represent unknown numbers.
• Recognizing that operations such as addition and subtraction are consistent even when represented by variables.
• Applying properties like the zero property to simplify expressions and solve equations.

By comprehending these basics, students are well-equipped to tackle more complex algebraic problems and apply these principles to both mathematical and real-world scenarios.