Problem 83 - Beginning and Intermediate Algebra

Question

Use the distributive property to rewrite each expression. $$ -0.6(8 x+1.2) $$

Step-by-Step Solution

Answer

-4.8x - 0.72

Step 1: Identify the Distributive Property

The distributive property states that for any numbers a, b, and c, the expression \( a(b + c) \) can be rewritten as \( ab + ac \). Here, we need to apply this to \( -0.6(8x + 1.2) \).

Step 2: Distribute \(-0.6\) to Each Term Inside the Parentheses

Multiply \(-0.6\) by each term inside the parentheses separately: \( -0.6 \times 8x \) and \( -0.6 \times 1.2 \).

Step 3: Calculate \(-0.6 \times 8x\)

Perform the multiplication: \( -0.6 \times 8x = -4.8x \).

Step 4: Calculate \(-0.6 \times 1.2\)

Perform the multiplication: \( -0.6 \times 1.2 = -0.72 \).

Step 5: Combine the Results

Combine the results of the two multiplications: \(-4.8x - 0.72\). Thus, \( -0.6(8x + 1.2) = -4.8x - 0.72 \).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify expressions and solve problems more easily. It states that for all real numbers a, b, and c, the following equation holds: \[a(b+c) = ab + ac\]. This property allows you to distribute a single term across terms within parentheses.
In our exercise, we have: \(-0.6(8x + 1.2)\).
We apply the distributive property by multiplying \(-0.6\) by each term inside the parentheses separately: \(-0.6 \times 8x\) and \(-0.6 \times 1.2\).
This step-by-step distribution helps break complex problems into smaller, more manageable parts.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators (such as addition, subtraction, multiplication, and division).
In our example, the expression inside the parentheses, \(8x + 1.2\), is an algebraic expression.
To rewrite this using the distributive property, we focus on how each term and operator interacts with other parts of the expression. It's essential to grasp these interactions to simplify and solve problems correctly.
The initial algebraic expression multiplies every term inside the parentheses by \(-0.6\), showing us how multiplication impacts each term involved.

Multiplication in Algebra

Multiplication is one of the core operations in algebra, and it often involves working with both numbers and variables. When using the distributive property, multiplication helps to simplify expressions into their component parts.
In our solution, we perform two multiplications:
1. \(-0.6 \times 8x = -4.8x\)
2. \(-0.6 \times 1.2 = -0.72\)
Combining these results, we rewrite the original expression as \(-4.8x - 0.72\).
Understanding how to work with multiplication, especially when combined with other operations, is crucial for solving algebraic expressions accurately.