Problem 72 - Beginning and Intermediate Algebra

Question

Use the distributive property to rewrite each expression. $$ 8(x-6) $$

Step-by-Step Solution

Answer

8x - 48

Step 1: Understand the Distributive Property

The distributive property states that for any numbers or expressions, a, b, and c, the expression \( a(b + c) = ab + ac \); similarly, \( a(b - c) = ab - ac \).

Step 2: Identify the Components

In the expression \( 8(x-6) \), identify a as 8, b as x, and c as 6.

Step 3: Apply the Distributive Property

Multiply 8 by both terms inside the parentheses: \( 8 \times x \) and \( 8 \times (-6) \).

Step 4: Perform the Multiplication

Calculate \( 8 \times x = 8x \) and \( 8 \times (-6) = -48 \).

Step 5: Combine the Results

The expression \( 8(x-6) \) distributes to \( 8x - 48 \).

Key Concepts

algebraic expressions

Algebraic expressions are a fundamental part of algebra. They consist of numbers, variables, and operations. In the given exercise, the algebraic expression is \(8(x-6)\). This means we have the number 8 outside the parentheses and a subtraction operation inside. Algebraic expressions can include:

• Variables like \(x\)
• Constants like 8 and -6
• Operations like multiplication and subtraction

Understanding how to manipulate and simplify these expressions helps solve equations and understand more complex math concepts. Remember, expressions do not have an equal sign; they are simply a combination of terms.

multiplication

Multiplication is an arithmetic operation that combines groups of equal sizes. In algebra, you multiply numbers and variables to simplify or expand expressions. Using the distributive property, multiplication involves distributing a term across terms within parentheses.
This means you multiply each term inside the parentheses by the term outside. For instance, in the expression \(8(x-6)\), you apply multiplication as follows:

• Multiply 8 by \(x\)
• Multiply 8 by -6

This gives us \(8 \times x = 8x\) and \(8 \times -6 = -48\). Multiplying carefully and keeping track of negative signs is crucial in algebra.

subtraction

Subtraction in algebra involves taking away one value from another. It is essential to keep track of the signs. In our example, \(8(x-6)\), we have a subtraction inside the parentheses. To handle this correctly with the distributive property, we must distribute 8 to both \(x\) and -6:

• First term: \(8 \times x\)
• Second term: \(8 \times -6\)

The minus sign before 6 means you are subtracting 8 times 6 from 8 times \(x\). This final operation simplifies to \(8x - 48\). Subtraction is about careful accounting of how much we take away, especially with multiple terms and variables.