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When working with rational equations, finding the least common denominator (LCD) is a crucial step. The LCD is the smallest expression or number that all denominators in a given equation can divide into without leaving a remainder. It's like finding a common ground for the denominators, so you can transform the entire equation into a solvable form.

In our exercise, the denominators are \(x\) and \(3\). Here’s how you find the LCD:

• List out the denominators: \(x\) and \(3\).
• Since \(x\) is a variable and \(3\) is a numerical constant, you will combine these two by multiplication.
• This yields \(3x\) as the least common denominator.

Having identified the LCD \(3x\) allows us to eliminate the fractions by multiplying every term in the equation by this LCD, making our calculations simpler and cleaner.

Once the rational equation is transformed using the least common denominator, you often end up with a linear equation. Linear equations are mathematical sentences where the highest power of the variable is one. They are straightforward to solve, typically involving basic operations such as addition, subtraction, multiplication, or division.

In this exercise, after multiplying the original rational equation by the LCD \(3x\), we received a linear equation: \(6 + x = 12\). Solving this kind of equation involves isolating the variable to one side of the equation.

• Start by subtracting \(6\) from both sides of the equation to maintain equality. This gives you \(x = 12 - 6\).
• Simplify the expression on the right-hand side, which results in \(x = 6\).

With linear equations, the goal is to have the variable by itself on one side of the equation, making it easy to read its value.

A step-by-step solution is a systematic method of breaking down the solution process into smaller, manageable parts. This approach helps in understanding each phase and dealing with potential complexities at each step.

Here is how you can systematically address this problem:

• Step 1: Find the least common denominator (LCD). In our equation, the LCD was \(3x\).
• Step 2: Multiply each term by the LCD \(3x\), which helped to eliminate the fractions: \(3x * 2/x + 3x * 1/3 = 3x * 4/x\), simplifying to \(6 + x = 12\).
• Step 3: Solve the linear equation: \(6 + x = 12\). Subtract \(6\) from each side to isolate \(x\), providing the solution \(x = 6\).

Follow these steps meticulously, and you'll find solving rational equations becomes an orderly and less daunting task. It's all about tackling one step at a time.