Problem 51 - Prealgebra

Question

Perform the indicated operations. $$4(3 x-2)$$

Step-by-Step Solution

Answer

The expression simplifies to \(12x - 8\).

Step 1: Distribute the 4

The expression given is \(4(3x-2)\). To expand it, you need to apply the distributive property, which states that \(a(b+c) = ab + ac\). Here, you will distribute the \(4\) to both terms inside the parentheses.

Step 2: Multiply 4 by 3x

Start by multiplying the number \(4\) with the term \(3x\). This gives you \(4 \times 3x = 12x\).

Step 3: Multiply 4 by -2

Now, multiply the \(4\) by the second term, \(-2\). This results in \(4 \times (-2) = -8\).

Step 4: Combine the results

Combine the results from the multiplication to complete the expression simplification. Therefore, the expanded expression is \(12x - 8\).

Key Concepts

Expression Simplification

Simplifying an expression often involves rewriting it in a clearer or more compact form. This process helps make complex problems easier to solve and understand. When dealing with an expression like \(4(3x - 2)\), simplification is done through the expansion. The key to expression simplification is to distribute or apply operations to each term within the parentheses.

• First, identify the operation inside the parentheses and determine how you can apply it to all terms.
• Next, simplify each term step by step using the correct arithmetic operations.
• Finally, combine all the simplified terms to write the final expression.

Through simplification, the expression gets transformed into a form that is much easier to work with, such as \(12x - 8\). This approach retains the equivalence of the expression while bringing clarity to the components involved. By practicing expression simplification, learners can gain confidence in handling more complex algebraic problems.

Multiplication in Algebra

Multiplication in algebra involves the application of basic multiplication principles to algebraic expressions. This is fundamental when expanding expressions, such as using the distributive property in the example \(4(3x - 2)\). The process requires understanding:

• Commutative property: In algebra, multiplication can be done in any order, \( ab = ba \).
• Associative property: Grouping of terms does not affect the result, \( (ab)c = a(bc) \).
• Distributive property: A core concept where a term is distributed across terms in parentheses: \( a(b + c) = ab + ac \).

In the given expression, these properties ensure each term is carefully multiplied individually by the coefficient outside the parentheses. This results in a new expression where each part is in its simplest form, as evidenced with the transformation to \(12x - 8\). Understanding these basic multiplication tenets helps manage larger expressions and equations efficiently.

Prealgebra Concepts

Prealgebra serves as the foundation for understanding algebraic operations and expressions. It lays out basic rules and operations needed to solve more complex mathematical problems in the future. In the expression \(4(3x - 2)\), core prealgebra concepts play a crucial role.These include:

• Understanding variables: Knowing \(x\) is an unknown but constant value that aims to be solved.
• Arithmetic operations: Mastery of basic arithmetic like addition, subtraction, multiplication, and division.
• Order of operations: Recognizing the correct sequence of operations, i.e., multiplication before addition/subtraction.

By grounding in prealgebra, learners frame how the parts of an algebraic expression interact and how operations are conducted. These foundational skills ensure a smooth transition into algebra, where expressions and equations become more involved. Comprehending prealgebra concepts solidifies an understanding of logical reasoning and prepares for higher-order thinking skills in mathematics.