Problem 82 - Beginning and Intermediate Algebra

Question

Use the distributive property to rewrite each expression. $$ -4(3 x-2) $$

Step-by-Step Solution

Answer

-12x + 8

Step 1: Recognize the Distributive Property

The distributive property states that for any numbers or expressions, \(a(b + c) = ab + ac\) and \(a(b - c) = ab - ac\). This means you can distribute the multiplication over addition or subtraction inside the parentheses.

Step 2: Apply the Distributive Property

To apply the distributive property, multiply \(-4\) by each term inside the parentheses: \(-4(3x)\) and \(-4(-2)\).

Step 3: Simplify the Expressions

Multiply the constants: \(-4 \times 3x = -12x\). Next, multiply \(-4 \times -2 = 8\). So the expression becomes \(-12x + 8\).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and operations. A variable represents an unknown value and is usually denoted by a letter like x or y. In the expression -4(3x - 2), -4 is a constant, 3x is a term with a variable, and -2 is also a constant. Learning to manipulate these expressions is essential in algebra. Always remember to follow the order of operations, typically known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Mastering Multiplication in Algebraic Context

Multiplication in algebra involves not just numbers but also variables and expressions. When using the distributive property, you multiply the term outside the parentheses by each term inside the parentheses individually. In the example -4(3x - 2), you have to distribute -4 to both 3x and -2. First, multiply constants with variables: -4 * 3x equals -12x. Then multiply the constants: -4 * -2 equals 8. These steps help simplify complex expressions and make solving equations easier.

Steps to Simplification

Simplification means reducing an expression to its simplest form. Use the distributive property to break down and simplify algebraic expressions. Here’s a quick recap:

• Distribute the multiplication across the terms inside the parentheses.
• Combine like terms if necessary.
• Ensure all calculations are correct, especially sign changes.

In the exercise -4(3x - 2), applying these steps results in -12x + 8. Simplifying expressions correctly ensures better understanding and easier problem-solving in algebra.