Solve For X: Missing Steps In Algebraic Equations

Algebra can sometimes feel like a detective story, where you're trying to uncover the value of an unknown, often represented by '$x$'. When we're faced with an equation like $2x - 6 = 4(x + 2)$, the goal is to isolate that '$x$' and find out what number it represents. This process involves a series of logical steps, and sometimes, like in our example, a couple of crucial moves might be missing. Let's dive into how we can figure out those missing steps and get to the solution, which in this case is '$x = -7$'. We'll be looking at how to correctly manipulate the equation to get there, ensuring we maintain balance throughout the process. It’s all about understanding the properties of equality – whatever you do to one side, you must do to the other!

Understanding the Equation and the Goal

Our starting point is the equation: $2x - 6 = 4(x + 2)$. The ultimate goal is to find the value of '$x$' that makes this statement true. To do this, we need to get all the terms with '$x$' on one side of the equation and all the constant terms (numbers without '$x$') on the other side. Think of the '=' sign as the center of a balanced scale. To keep the scale balanced, any operation we perform on one side must be mirrored on the other. The provided solution, '$x = -7$', is our target. We need to ensure our steps lead us logically and correctly to this answer. Let's look at the steps that are given and identify where the gaps are. We have the initial equation, then a simplified form, and finally, the solution. The simplification step, $2x - 6 = 4x + 8$, is where the first action of distributing the '4' across the terms inside the parentheses has already occurred. So, the first transformation from $2x - 6 = 4(x + 2)$ to $2x - 6 = 4x + 8$ correctly applies the distributive property. Now, the real work of isolating '$x$' begins, and this is where the missing steps usually lie.

The First Missing Step: Consolidating the '$x$' Terms

After the initial distribution, we have $2x - 6 = 4x + 8$. Notice that we have '$x$' terms on both sides of the equation. To solve for '$x$', we need to bring all the '$x$' terms together. A common strategy is to move the '$x$' term with the smaller coefficient to the side with the '$x$' term with the larger coefficient. In this case, '$2x$' is smaller than '$4x$'. To eliminate '$2x$' from the left side, we perform the inverse operation: subtract '$2x$' from both sides. This maintains the equality of the equation. So, the step would look like this: $2x - 6 - 2x = 4x + 8 - 2x$. Simplifying this gives us $-6 = 2x + 8$. Alternatively, we could choose to move the '$4x$' term to the left side by subtracting '$4x$' from both sides. This would look like: $2x - 6 - 4x = 4x + 8 - 4x$. Simplifying this yields $-2x - 6 = 8$. Both approaches are valid, but let's stick with the first one where we end up with a positive coefficient for '$x$' for simplicity. The key here is that the addition of the $4x$ variable from one side to move the $x$ to one side is a slightly confusing description of the operation. More accurately, it's the subtraction of '$2x$' (or '$4x$') from both sides to consolidate the '$x$' terms. The phrasing in option A,