Problem 41 - Prealgebra

Question

Apply the distributive property to each expression and then simplify. $$2(x+3)+4(x+2)$$

Step-by-Step Solution

Answer

The simplified expression is \(6x + 14\).

Step 1: Distribute 2 in the first expression

Apply the distributive property by multiplying 2 with each term inside the parentheses in the expression \(2(x+3)\). This gives us \(2 \cdot x + 2 \cdot 3 = 2x + 6\).

Step 2: Distribute 4 in the second expression

Use the distributive property to multiply 4 with each term inside the parentheses in the expression \(4(x+2)\). This results in \(4 \cdot x + 4 \cdot 2 = 4x + 8\).

Step 3: Combine like terms

Now, add the expressions from Step 1 and Step 2 together: \((2x + 6) + (4x + 8)\). Combine the like terms by adding \(2x + 4x\) and \(6 + 8\), which simplifies to \(6x + 14\).

Key Concepts

Simplifying Expressions

When you're working with algebraic expressions, one core concept you'll encounter is simplifying expressions. This often involves using the distributive property, which is all about removing parentheses by distributing a factor across terms inside the parentheses. For instance, let's start with the expression \(2(x + 3)\). By distributing the \(2\), you multiply it by \(x\) and \(3\), giving you \(2x + 6\). This simplifies the expression by getting rid of the parentheses.

Simplifying expressions is a fundamental skill because it makes equations easier to solve and understand. Once you apply the distributive property and remove the parentheses, you then focus on combining like terms, which further simplifies the equation. This step-by-step breakdown helps not just in simplifying but also in building a solid foundation in algebra for more complex operations.

Combine Like Terms

After applying the distributive property to eliminate the parentheses, the next big step in simplification is to combine like terms. But what are like terms? They are terms in the expression that have the same variable raised to the same power, such as \(2x\) and \(4x\) or plain numbers like \(6\) and \(8\).

- Start by identifying all of the like terms in an expression. - Then, add or subtract the coefficients of these terms.
In our example, once you distribute, you are left with \(2x + 6 + 4x + 8\). Combining the like terms \(2x\) and \(4x\), you get \(6x\). Similarly, adding the constants, \(6 + 8\), gives you \(14\). This transforms our expression into \(6x + 14\), which is much simpler and clearer.

By combining like terms correctly, we achieve a tidier expression that is easier to work with in further mathematical operations or problems.

Prealgebra Math Exercises

In prealgebra, exercises like simplifying expressions are common and serve as a building block for more advanced algebra concepts. They provide critical practice in using the distributive property and combining like terms, which are both foundational elements in algebra. These exercises lay the groundwork for topics students will encounter later, such as solving equations and working with polynomials.

Prealgebra exercises often include:

• Practicing the application of the distributive property
• Identifying and combining like terms
• Solving basic algebraic equations

Working through these problems also helps improve mathematical thinking and problem-solving skills. By repeatedly applying concepts like the distributive property, students gain confidence and skill in handling more complex algebraic tasks that will come later in their educational journey.