Problem 383 - Intermediate Algebra

Question

Simplify \(8\left(x-\frac{1}{4}\right)\) using the Distributive Property and explain each step.

Step-by-Step Solution

Answer

The simplified expression is \(8x - 2\).

Step 1: Identify the Expression

The given expression is \(8\left(x - \frac{1}{4} \right)\). We need to simplify this expression using the Distributive Property.

Step 2: Apply the Distributive Property

Using the Distributive Property, multiply 8 with each term inside the parentheses: \[8 \times x - 8 \times \frac{1}{4}\]. This will break the expression into two parts.

Step 3: Simplify Each Term

Simplify the two parts separately: \[8 \times x = 8x\] and \[8 \times \frac{1}{4} = 2\].

Step 4: Combine Simplified Terms

Combine the simplified terms to get the final expression: \(8x - 2\).

Key Concepts

Simplifying Expressions

Simplifying expressions is all about reducing a mathematical expression to its simplest form. This often involves combining like terms or applying specific properties, such as the Distributive Property. Simplifying helps in solving equations more efficiently and makes complex problems more manageable. It's essential to understand the idea behind each step to ensure you can apply it correctly in different scenarios.

Algebraic Operations

Algebraic operations include addition, subtraction, multiplication, and division performed on algebraic expressions. Let's break down the operations used in the given example:

The original expression is \( 8(x - \frac{1}{4}) \). To simplify it, we use the Distributive Property, which states that for any numbers a, b, and c, \( a(b + c) = ab + ac \). Here, we multiply 8 by each term inside the parentheses:

• First term: \( 8 \times x = 8x \)
• Second term: \( 8 \times \frac{1}{4} = 2 \)

After performing these operations separately, we combine the results to form the final simplified expression: \( 8x - 2 \).

Step-by-Step Solutions

Understanding step-by-step solutions is crucial because it breaks down a problem into smaller, more digestible parts. Here's a detailed explanation of the steps we took to simplify the expression \( 8(x - \frac{1}{4}) \):

• Step 1: Identify the Expression
  The given expression is \( 8(x - \frac{1}{4}) \). Recognize that we need to use the Distributive Property to simplify it.
• Step 2: Apply the Distributive Property
  Multiply 8 with each term inside the parentheses. This gives us: \( 8 \times x - 8 \times \frac{1}{4} \).
• Step 3: Simplify Each Term
  Perform the multiplications separately: \( 8 \times x = 8x \) and \( 8 \times \frac{1}{4} = 2 \).
• Step 4: Combine Simplified Terms
  Finally, add the simplified terms together to get: \( 8x - 2 \).

By following these steps, you can systematically simplify many similar algebraic expressions. Practice this technique to gain confidence and proficiency in algebra.