Solving the Equation 4(3x - 2) = 2(3x + 8)

The problem asks us to find the value of \(x\) that satisfies the linear equation \(4(3x - 2) = 2(3x + 8)\). To solve this equation, we need to use algebraic properties to isolate the variable \(x\) on one side of the equation.

Let's break down the solution step-by-step:

• Step 1: Apply the Distributive Property.
  We need to multiply the number outside the parentheses by each term inside the parentheses on both sides of the equation. \[4(3x - 2) = 4 \times 3x - 4 \times 2 = 12x - 8\] \[2(3x + 8) = 2 \times 3x + 2 \times 8 = 6x + 16\] So, the equation becomes: \[12x - 8 = 6x + 16\]
• Step 2: Gather the \(x\) terms on one side.
  To do this, we can subtract \(6x\) from both sides of the equation to move the \(x\) terms to the left side. \[12x - 8 - 6x = 6x + 16 - 6x\] \[(12x - 6x) - 8 = (6x - 6x) + 16\] \[6x - 8 = 16\]
• Step 3: Gather the constant terms on the other side.
  To do this, we can add \(8\) to both sides of the equation to move the constant terms to the right side. \[6x - 8 + 8 = 16 + 8\] \[6x = 24\]
• Step 4: Isolate \(x\).
  The term \(6x\) means \(6\) multiplied by \(x\). To isolate \(x\), we need to divide both sides of the equation by \(6\). \[\frac{6x}{6} = \frac{24}{6}\] \[x = 4\]

The value of \(x\) that satisfies the given equation is \(4\).

Comparing our result to the given options:

• A. 1
• B. 2
• C. 3
• D. 4

Our calculated value \(x = 4\) matches option D.

Therefore, the correct answer is \(x = 4\).

Additional Information: Linear Equations and Their Properties

A linear equation in one variable is an equation that can be written in the standard form \(Ax + B = 0\), where \(A\) and \(B\) are constants and \(A \neq 0\). The equation we solved, \(4(3x - 2) = 2(3x + 8)\), is a linear equation because it simplifies to the form \(6x - 24 = 0\), where \(A=6\) and \(B=-24\).

Solving linear equations involves performing inverse operations to isolate the variable. The goal is to get the variable term on one side and the constant terms on the other.

• Operations like addition and subtraction are inverse operations.
• Operations like multiplication and division are inverse operations.

Whatever operation you perform on one side of the equation, you must perform the same operation on the other side to keep the equation balanced. This is the fundamental principle behind solving equations using the properties of equality.