Problem 68 - Algebra 1

Question

Apply the distributive property. $$4(x+2)$$

Step-by-Step Solution

Answer

The result using the distributive property is \(4x + 8\).

Step 1: Identify the factor and the terms within parentheses

In this exercise, the factor is 4 and the terms inside the parentheses are x and 2.

Step 2: Apply distributive property

The distributive property states that a(b + c) = ab + ac. Applying this property to our exercise gives us: 4x + 4 * 2.

Step 3: Multiply the factor with each term

4 times x is 4x and 4 times 2 is 8. Therefore, it would become 4x + 8.

Key Concepts

Understanding Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. It allows us to solve problems by expressing relationships between quantities that can vary. In algebra, these symbols (often represented as letters like \( x \), \( y \), or \( z \)) stand in for numbers whose values we either do not know yet or can be any number in a specific problem.

The ability to manipulate these symbols and equations gives you powerful tools to solve a variety of real-world problems. It allows you to set up equations and solve for unknowns in situations ranging from basic arithmetic to complex systems.

Here are a few key points to remember about algebra:

• Algebraic expressions are combinations of variables, numbers, and operations.
• Equations are statements that two expressions are equal and often require solving for a variable.
• Being familiar with the properties and operations in algebra is crucial for solving complex equations.

The Role of Multiplication

Multiplication is one of the fundamental operations in mathematics. It is essentially repeated addition. For example, multiplying 4 by 3 simply means adding 4 three times: \( 4 + 4 + 4 \).

In algebra, multiplication plays a crucial role in expressing relationships and simplifying expressions. It can be used to scale terms, solve equations, and simplify complex expressions.

When applying the distributive property, multiplication allows you to "distribute" or spread out a factor over terms inside parentheses, as shown in the exercise:

• The expression \( 4(x + 2) \) is simplified by multiplying 4 with both \( x \) and 2.
• This yields the expression \( 4x + 8 \), showing how multiplication distributes over addition.

Understanding how multiplication interacts with other operations is vital for mastering algebraic concepts.

Working with Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operations that represent a particular value or set of values. These expressions are tools used to describe relationships and quantities without necessarily solving them.

In algebra, working with expressions involves simplifying and manipulating these combinations to make them more understandable or useful. The distributive property, for instance, is a technique used to break down expressions into simpler parts, making them easier to solve or simplify.

Considerations when manipulating mathematical expressions include:

• Identifying like terms, which are terms with the same variable raised to the same power, that can be combined.
• Using algebraic rules and properties, like the distributive property, to simplify expressions.
• Recognizing and applying arithmetic operations, such as addition or subtraction, to combine terms.

By mastering these techniques, students gain the ability to work with increasingly complex algebraic expressions and equations.