Problem 48 - College Algebra with Modeling and Visualization

Question

Apply the distributive property. $$4(x-7)$$

Step-by-Step Solution

Answer

The expanded expression is \(4x - 28\).

Step 1: Identify the Multiplicands

Look at the expression given: \(4(x-7)\). Here, 4 is the number outside the parenthesis, and \((x-7)\) is the expression inside the parenthesis. The distributive property allows us to multiply the number outside the parenthesis by each term within the parenthesis.

Step 2: Apply the Distributive Property

To apply the distributive property in \(4(x-7)\), we need to multiply 4 by each term inside the parenthesis individually. This means we will multiply 4 by \(x\) and 4 by \(-7\).

Step 3: Multiply Each Term

Perform the multiplication: 1. Multiply 4 by \(x\): \(4 imes x = 4x\). 2. Multiply 4 by \(-7\): \(4 imes -7 = -28\).

Step 4: Combine the Results

Now, combine the results of the multiplications: The expression becomes \(4x - 28\). This is the expanded form of the original expression using the distributive property.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. These expressions are written using letters to represent unknown values, which allows us to solve problems with varied inputs. For example, in the expression \(4(x-7)\), the letter \(x\) is a variable, representing a value that can change.

It's important to understand that every algebraic expression has distinct components:

• **Coefficients**: These are numbers that are multiplied by the variables (e.g., the number 4 in \(4x\)).
• **Variables**: Symbols like \(x\) or \(y\) that represent unknown values.
• **Constants**: Fixed numbers that do not change, like 7 in the expression \(x-7\).

By understanding the parts of an algebraic expression, you can better manipulate and simplify them to make solving problems easier.

Multiplicative Operations

Multiplicative operations, as seen in algebra, involve the process of multiplying components within an expression or equation. This process is crucial when applying the distributive property: a method used to simplify expressions like \(4(x-7)\).

The distributive property is a fundamental algebraic rule that states:

• **Distribute One Term Over Others**: If you have a term outside a parenthesis, like 4 in our example, multiply it with each term inside the parenthesis individually.

For the expression \(4(x-7)\), follow these steps:

• Multiply 4 by \(x\) to get \(4x\).
• Multiply 4 by \(-7\) to get \(-28\).

These operations demonstrate the application of multiplicative operations, turning a more complex expression into something simpler that can be analyzed or solved easily.

Simplifying Expressions

Simplifying expressions is a vital process in algebra that makes complex expressions easier to understand and solve. When you simplify an expression, you re-write it into an equivalent, but easier-to-handle form. This often involves using algebraic properties such as the distributive property.

In our example of \(4(x-7)\), applying the distributive property simplified the expression to \(4x - 28\). Here's how it works:

• Start by applying the distributive property, multiplying each term inside the parentheses by the factor outside.
• Combine like terms, if there are any. In this case, no like terms exist beyond what's already calculated, so the expression is already as simple as it can get.

Simplifying makes solving equations and algebraic expressions much easier, allowing you to efficiently handle potentially complex mathematical problems by breaking them into straightforward steps.