Problem 49 - Algebra A Combined Function

Question

Remove parentheses and simplify each expression. $$ 2+4(6 x-6) $$

Step-by-Step Solution

Answer

The simplified expression is \(24x - 22\).

Step 1: Distribute the 4

Start by distributing the 4 in the expression \(4(6x - 6)\). This means you multiply 4 by each term inside the parentheses: \(4 \times 6x = 24x\) and \(4 \times (-6) = -24\). So, \(4(6x - 6)\) becomes \(24x - 24\).

Step 2: Combine Like Terms

Take the expression from Step 1 and combine it with any remaining terms. The entire expression is now \(2 + 24x - 24\). Now, combine the constant terms \(2\) and \(-24\), which results in \(-22\). So, the expression simplifies to \(24x - 22\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps us simplify expressions by removing parentheses. It states that when you multiply a number by a sum or difference inside parentheses, you distribute the multiplication over each term. In simpler terms, you multiply the number outside the parentheses by each term inside the parentheses individually. In the given expression, you have to distribute the 4 across the terms within the parentheses: \(4(6x - 6)\).

• Step 1: Multiply 4 by \(6x\), giving you \(24x\).
• Step 2: Multiply 4 by \(-6\), resulting in \(-24\).

Ultimately, when 4 is distributed, the expression \(4(6x - 6)\) simplifies to \(24x - 24\). Distributing ensures the parentheses are eliminated, allowing us to move on to simplification.

Combining Like Terms

Combining like terms is the process of merging terms that have the same variable raised to the same power. It makes algebraic expressions simpler and easier to work with. In our expression, after applying the distributive property, we have \(2 + 24x - 24\). The goal here is to find terms with the same variable and constant terms and combine them.

• Step 1: Identify like terms. In this expression, \(24x\) is a term with a variable, and \(2\) and \(-24\) are constant terms.
• Step 2: Combine the constant terms. Add \(2\) and \(-24\), which gives you \(-22\).

This results in the simplified expression \(24x - 22\). Always ensure that only like terms are combined to maintain the integrity of the expression.

Parentheses Removal

Removing parentheses is an integral part of solving and simplifying algebraic expressions. Parentheses group parts of expressions together and dictate the order of operations. By applying the distributive property, we eliminate these parentheses, making the expression easier to handle.

To remove parentheses, you multiply each term inside the parentheses by the term outside the parentheses. Let's revisit our expression: \(2 + 4(6x - 6)\). Once the distributive property is applied, it becomes \(2 + 24x - 24\), and the parentheses are removed.

• Ensure each term in the parentheses is multiplied by the outside number to eliminate the parentheses.
• Once the expression is free of parentheses, proceed to combine like terms for further simplification.

Successfully removing parentheses sets the stage for a clear and straightforward simplification process, leading us to a solution that is complete and easy to understand: \(24x - 22\).