Problem 33 - Prealgebra

Question

Use the Distributive Property to write each expression as an equivalent algebraic expression. $$(x+3) 4$$

Step-by-Step Solution

Answer

The expression simplifies to \(4x + 12\).

Step 1: Identify the Expression

We need to use the Distributive Property to simplify the expression \((x+3) 4\). The expression \(x + 3\) and the number 4 are the key components here.

Step 2: Apply the Distributive Property

The Distributive Property allows us to multiply a number by each term inside the parentheses. For the expression \((x+3) 4\), we distribute the 4 to both \(x\) and 3.

Step 3: Multiply Each Term

First, multiply 4 by \(x\) to get \(4x\). Next, multiply 4 by 3 to get 12. The results of these multiplications combine to form the new expression.

Step 4: Combine the Results

Combine the results from Step 3 to write the equivalent expression: \(4x + 12\). This is the expression simplified using the Distributive Property.

Key Concepts

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and operations. They often look like math sentences that can contain constants (fixed numbers), variables (symbols that can change or take on various values), and operations like addition, subtraction, multiplication, and division. Consider

• Example: In the expression \( x + 3 \), \( x \) is a variable, and 3 is a constant.
• Applications: These are foundational in algebra and lay the groundwork for creating equations and inequalities.

Understanding how these expressions are composed is crucial when you begin to manipulate them, such as when applying operations like the Distributive Property. This property allows us to transform and simplify expressions, aiding in solving whether you need to balance an equation or interpret a problem situation. Begin by identifying each part of the expression and how it interacts with other parts; it simplifies future manipulations.

Simplifying Expressions

Simplifying expressions involves rewriting them in a more straightforward or efficient form. It helps make calculations easier and the expressions easier to understand. To simplify the expression

• Identify: Look at the expression to understand what operations are needed.
• Use properties: Apply properties like the Distributive Property appropriately to break down and reassemble parts of the expression.

For example, starting with \((x+3) 4\), we apply the Distributive Property to get \(4x + 12\). Simplifying involves:

• Distributing: Multiplies each term within the parenthesis by the outside number.
• Combining like terms: Merges terms with the same variable parts to loosen clutter.

Through simplification, you reduce the opportunity for mistakes in evaluating expressions, paving the way toward easier handling of complex calculations.

Multiplying Terms

Multiplying terms in expressions, like using the Distributive Property, involves multiplying a single term by multiple terms inside parentheses. In

• Process: The outer term multiplies into each component inside the brackets.
• Example: Consider the expression \((x+3) 4\).

The multiplication process involves:

• Multiplying \(4\) by \(x\), resulting in \(4x\).
• Multiplying \(4\) by \(3\), resulting in \(12\).
• Combining these results to form the simplified expression \(4x + 12\).
• Ensure arithmetic accuracy: Verify each multiplication is performed correctly.

This technique is a vital skill as it is used frequently in algebra to transform and solve equations. The better you understand multiplying across terms, the more fluidly you will handle more complicated algebraic tasks.