Problem 50 - Algebra A Combined Function

Question

Remove parentheses and simplify each expression. $$ 8+4(3 x-4) $$

Step-by-Step Solution

Answer

The simplified expression is \(12x - 8\).

Step 1: Distribute 4

To remove the parentheses, distribute the 4 to each term inside the parentheses by multiplying it with each one. The expression inside the parentheses is \(3x-4\), so multiply both terms by 4: - \(4 \times 3x = 12x\) - \(4 \times -4 = -16\) This results in the expression \(8 + 12x - 16\).

Step 2: Combine Like Terms

Now combine any like terms to simplify the expression. The like terms here are the constants 8 and -16. - Add these constants together: \(8 - 16 = -8\)Therefore, the expression simplifies to \(12x - 8\).

Key Concepts

Removing Parentheses

When simplifying expressions, one of the first tasks is to remove the parentheses. This step is crucial because it allows you to work with individual terms and helps with further simplification. To remove parentheses effectively, you often use the Distributive Property, which we'll discuss in the next section.

For the expression provided, \[8 + 4(3x - 4),\]removing the parentheses involves making sure each term inside the parentheses is dealt with properly. Without doing this, you can't combine or rearrange terms accurately.

Typically, removing parentheses involves multiplying the term outside the parentheses by each term inside. This ensures that the parentheses no longer affect the expression, making it easier to simplify.

Distributive Property

The Distributive Property is one of the foundational principles in algebra. It's used to eliminate parentheses and distribute a term across an addition or subtraction inside parentheses. The formal statement of this property is:\[a(b + c) = ab + ac\]This means you multiply each term inside the parentheses by the term outside.

Let's see how this applies to our expression \[8 + 4(3x - 4).\]The term outside the parentheses is 4. We apply it to both terms inside:

• First, multiply 4 with the first term inside: \(4 \times 3x = 12x\)
• Then, multiply 4 with the second term: \(4 \times -4 = -16\)

After distribution, the expression becomes \[8 + 12x - 16.\]By distributing effectively, we've made it possible to further simplify by combining like terms, which we'll discuss next.

Combining Like Terms

Once you've managed to remove the parentheses using the Distributive Property, it's time to simplify the expression further by combining like terms. Like terms are terms that have the same variable raised to the same power, or constant terms without any variables.

In our expression\[8 + 12x - 16,\]you have constants (8 and -16) and a term with the variable (12x). Combining like terms means adding or subtracting these constants:

• Combine the constants: \(8 - 16 = -8\)

Now, the expression simplifies to:\[12x - 8.\]Congratulations! You have simplified your expression fully. This practice helps ensure that expressions are in their simplest form, making them easier to work with in subsequent calculations or equations.