Problem 75 - Beginning and Intermediate Algebra

Question

Use the distributive property to rewrite each expression. $$ -\frac{1}{4}(8 x+3) $$

Step-by-Step Solution

Answer

-2x - \frac{3}{4}

Step 1: Identify the distributive property

The distributive property states that for any numbers a, b, and c, the expression can be expanded to .Identify the term outside the parenthesis and the terms inside.

Step 2: Multiply the outer term with the first term inside the parenthesis

The term outside the parenthesis is . First, distribute by multiplying : .

Step 3: Multiply the outer term with the second term inside the parenthesis

Next, distribute by multiplying : .

Step 4: Combine the results

Now combine the results from steps 2 and 3 to write the final expression: .

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables (like \(x\) or \(y\)), and operational symbols (such as +, -, *, and /). For example, in the expression \(8x + 3\), \(8x\) and \(3\) are the terms. Variables are symbols that represent numbers, and they can change value depending on the situation. The goal is often to simplify or solve these expressions.

When working with algebraic expressions, you may need to use several mathematical properties and rules. One crucial rule is the distributive property, which helps in breaking down complex expressions into simpler parts.

Exploring the Properties of Operations

The properties of operations are fundamental rules that make it easier to perform calculations. These include:

• Commutative Property: Order doesn't matter in addition or multiplication, e.g., \(a + b = b + a\) or \(a \times b = b \times a\).
• Associative Property: Grouping doesn't matter in addition or multiplication, e.g., \((a + b) + c = a + (b + c)\) or \((a \times b) \times c = a \times (b \times c)\).
• Distributive Property: Multiplication distributes over addition or subtraction, e.g., \(a(b + c) = ab + ac\).

The distributive property is particularly useful for expanding algebraic expressions. For instance, in the expression \(-\frac{1}{4}(8x + 3)\), we apply this property to multiply everything inside the parentheses by \(-\frac{1}{4}\).

Mastering Multiplication in Distributive Property

To fully understand the distributive property, it's crucial to focus on the multiplication involved. Let's break down the example given: \(-\frac{1}{4}(8x + 3)\).

Step 1: Identify the Components
Here, the outer term is \(-\frac{1}{4}\), and the inner terms are \(8x\) and \(3\).

Step 2: Distribute and Multiply
First, multiply the outer term by the first inner term: \(-\frac{1}{4} \times 8x = -2x\).
Next, multiply the outer term by the second inner term: \(-\frac{1}{4} \times 3 = -\frac{3}{4}\).

Step 3: Combine the Products
Summing these results, the expression simplifies to \(-2x - \frac{3}{4}\).

Remember to carefully follow the steps and apply the distributive property consistently to achieve accurate results.