Problem 67 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(2(4 x+6)+3\)

Step-by-Step Solution

Answer

The simplified expression is \[8x + 15\].

Step 1: Identify the expression to simplify

We begin with the expression: \[2(4x + 6) + 3\]

Step 1: Distribute the constant outside the parenthesis

Distribute the 2 to both terms inside the parentheses. This involves multiplying 2 by each term within the parenthesis:\[2(4x) + 2(6)\]

Step 2: Perform multiplication

Now, calculate the products:\[2 \times 4x = 8x\] \[2 \times 6 = 12\]This transforms our expression into:\[8x + 12 + 3\]

Step 3: Combine like terms

Combine the constants (12 and 3) to simplify the expression further:\[8x + (12 + 3)\]This simplifies to:\[8x + 15\]

Key Concepts

The Distributive Property

The distributive property is a fundamental algebraic principle that allows you to simplify expressions that involve multiplication over addition or subtraction. It's a helpful tool in many algebra problems.
To distribute means to multiply a single term outside the parentheses by each term inside the parentheses. For instance, in the expression \(2(4x + 6)\), the number 2 outside the parentheses must be multiplied by both \(4x\) and \(6\).
Here is how the distributive property works in this problem:

• First, we distribute the 2 by multiplying it with \(4x\) and \(6\).
• We get 2 \(\cdot 4x + 2 \cdot 6 \).
• This simplifies to \(8x + 12\).

Using the distributive property helps break down complex expressions into simpler parts, making it easier to solve.

Combining Like Terms

Combining like terms is another crucial step in simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power, though their coefficients might be different.
For instance, \(8x\) and \(12\) are not like terms because \(8x\) has a variable part \(x\) whereas 12 is just a constant.
After using the distributive property, the expression becomes \(8x + 12 + 3\). To simplify further, we combine the constants 12 and 3:

• Identify like terms: Here, \(12\) and \(3\) are like terms because they are both constants.
• Add them together: \(12 + 3 = 15\).

This gives us the simplified expression \(8x + 15\). By combining like terms, we reduce the complexity of the expression.

Multiplication in Algebra

Multiplication in algebra mostly involves multiplying constants with variables or other constants. Understanding multiplication helps in steps like using the distributive property.
In the given problem, we see multiplication applied within the distributive property:

• 2 \(\cdot\) \(4x\) = \(8x\).
• 2 \(\cdot\) \(6\) = \(12\).

Multiplication in algebra follows the same rules as basic arithmetic but adds the complexity of variables. Remember that when you multiply a constant with a variable, you just place the constant next to the variable. For example, \(2 \cdot 4x = 8x\), as shown in our problem. By mastering multiplication, you can easily handle more complex algebraic expressions.