Problem 91 - Introductory Algebra for College Students

Question

Simplify each algebraic expression. $$-3(-2 x+4)$$

Step-by-Step Solution

Answer

The simplest form of \(-3(-2x+4)\) is \(6x - 12\).

Step 1: Distribution

First, distribute -3 to each term inside the parentheses. Therefore, \(-3(-2x+4)\) simplifies to \(6x - 12\).

Step 2: Simplify the Expression

There are no like terms to combine in the expression \(6x - 12\), so this is the simplest form of the given expression.

Key Concepts

distribution in algebra

When you encounter an algebraic expression like \(-3(-2x + 4)\), one of the first steps to simplifying it is by using distribution. Distribution in algebra involves multiplying each term within the parentheses by the factor outside of it. This is known as the distributive property. The distributive property states that \(a(b + c) = ab + ac\).
For our example, \(-3(-2x + 4)\), \(-3\) is the factor outside the parentheses. To distribute \(-3\) through the expression, you need to:

• Multiply \(-3\) by each term inside the parentheses separately.
• Perform the multiplications: \(-3 \times -2x = 6x\) and \(-3 \times 4 = -12\).

This transforms the expression to \(6x - 12\), and the distribution is complete.

algebraic expressions

In algebra, expressions are made up of numbers, variables, and operation symbols all combined together. An expression can be as simple as a single number or as complicated as formulas involving multiple terms. Let's break down the term 'algebraic expression' with a focus on its essential components:
An algebraic expression such as \(-3(-2x + 4)\):

• Contains numbers (constants or coefficients), such as \(-3\) and \(4\).
• Involves a variable, which in this case is \(x\).
• Incorporates arithmetic operations like addition, subtraction, and multiplication.

These components are critical as they allow us to form expressions that can be manipulated and solved using algebraic methods, making it essential to understand each part's role within the expression.

simplifying expressions

Simplifying an algebraic expression is the process of making it as concise as possible without altering its value. After performing distribution, you may end up with terms that can’t be combined because they are not alike. In our exercise, \(6x - 12\), there are no like terms to combine, and thus it is already in its simplest form.
Here are some general steps to simplify an expression:

• First, perform any operations inside parentheses using the distributive property if necessary.
• Next, combine like terms. Like terms have the same variable raised to the same power.
• Finally, ensure no further simplification is possible.

This process not only shortens the expression but also clarifies and prepares it for further operations if necessary. Remember, the goal is to make the expression as straightforward as possible while maintaining its initial meaning.