Solving 3(x - 4) = 12: a straightforward way to find x

Cracking 3(x - 4) = 12: a friendly road map to a clean solution

If you’ve ever stared at a sentence like 3(x - 4) = 12 and felt a little tug of confusion, you’re in good company. Equations can look intimidating when they hide the variable inside parentheses, but there’s a simple rhythm you can follow. Think of the equal sign as a balance beam and each operation as something you add to or remove from one side to keep the balance intact. The goal is to isolate x, and you’ll see it isn’t as scary as it first appears.

What the problem is up to

Here’s the setup you’re given:

3(x - 4) = 12

Multiple-choice options are A) 0, B) 4, C) 8, D) 12. The correct answer is 8. But let’s walk through why that is, so you feel confident not just with this problem, but with similar ones that show up on the HSPT.

Step 1: Unwrap the outer layer (divide to isolate the inner chunk)

The key idea is to treat the whole thing inside the parentheses, (x - 4), as a single unit that’s being multiplied by 3. Since 3 times something equals 12, that “something” must be 12 divided by 3, which is 4. In equation form:

3(x - 4) = 12

Divide both sides by 3: x - 4 = 4

Why this works is simple: division is the inverse of multiplication. If 3 times a value gives you 12, that value is 12/3, which is 4. Then you’re left with the neat inside part, x - 4, sitting alone on one side.

Step 2: Undo the minus 4 to isolate x

Now you have x - 4 = 4. The next move is to undo the subtraction of 4, which means adding 4 to both sides. The goal is to land x by itself.

x - 4 = 4

Add 4 to both sides: x = 4 + 4

And that gives you:

x = 8

Simple, right? The arithmetic is clean, the logic is solid, and you can see why this is the most direct path to the answer.

Step 3: Quick sanity check (always a good habit)

To confirm, substitute x = 8 back into the original equation:

3(8 - 4) = 3(4) = 12

Both sides match, so x = 8 is correct. A quick check like this helps catch slips before you move on to the next problem.

Why this approach makes sense (and how to internalize it)

Two simple principles often show up when you’re solving linear equations:

• Isolate the variable by undoing operations in reverse order.

• Treat parentheses as one unit. If a number multiplies a whole expression, you can undo that multiplication by dividing the entire expression on the same side.

In this problem, the first move was to isolate the inner expression by dividing by 3, which turns the left side into x - 4. The second move then isolates x by undoing the subtraction inside the parentheses. It’s a tidy sequence: divide, then add.

A few friendly reminders to keep when you face similar problems

• Remember what the inverse operations are: multiplication ↔ division, addition ↔ subtraction. If you can map out which operation you’re undoing, you can chart a clear path from the given equation to x.

• Keep an eye on the parentheses. The number in front (the 3) is multiplying everything inside (x - 4). Don’t try to apply the operation to just part of the expression. Treat the whole parenthetical as one unit.

• If you expand first, you’ll still reach the same answer sometimes, but expanding can blur the structure of the problem. It’s usually cleaner to undo the outer operation first, then simplify inside.

A small digression that ties it to real life

Think of this like balancing a shopping bill. If you’re told that three people reimburse you for a shared item, and the total is 12 dollars, you’d first figure out how much each person contributed by dividing the total by 3. Then, if you know one person’s share is the amount in parentheses (the “x minus 4” part), you’d adjust to find the exact contribution. It’s the same math idea—break a big thing into its parts, then work backward to the unknown.

Common pitfalls (and how to sidestep them)

• Skipping the division step: It’s tempting to try to distribute first, but you can still arrive at the right answer by distributing, you’ll see the same result (3x - 12 = 12 → 3x = 24 → x = 8). The clean route is often dividing first to keep the structure clear.

• Forgetting to apply the same operation to both sides: When you divide both sides by 3, you must do so on both sides. If you only adjust one side, you’ll misbalance the equation.

• Overcomplicating it with algebraic gymnastics: This problem is a textbook case of inverse operations. Overthinking can fog the simple logic that’s right in front of you.

A couple of quick variations to try in your head

If you want to get a feel for the same pattern, try these in a zero-pressure way:

• Solve 2(y + 6) = 20. First divide both sides by 2: y + 6 = 10. Then subtract 6: y = 4.

• Solve 5(z - 1) = 25. Divide by 5: z - 1 = 5. Add 1: z = 6.

• Solve 7(w - 2) = 21. Divide by 7: w - 2 = 3. Add 2: w = 5.

Notice how the steps line up with the original problem: divide to peel off the multiplier, then adjust inside the parentheses to isolate the variable.

A mindset that helps more than one kind of algebra problem

• Start with the goal in mind: you want x by itself.

• Look for the single, biggest operation acting on the whole variable expression.

• Undo that operation first, using the inverse operation, and keep the balance intact.

• Then tidy up any inner expressions to finish.

If you’re the kind of learner who likes a small mental script, try this rhyme: “Divide to reveal, then add to seal.” It’s not fancy, but it sticks, and it can spare you those moments of hesitation at the whiteboard.

Putting it all together

We started with 3(x - 4) = 12 and landed on x = 8, with a clean check to boot. That’s a solid demonstration of the core idea behind many linear equations: identify the outer operation, undo it, then deal with the inside of the parentheses. The logic is consistent, the steps are straightforward, and you can apply the exact same pattern to a wide range of problems you’ll encounter on the test.

A small anthology of hints you can carry around

• Treat parentheses as a single block. If a number multiplies the whole block, divide to undo.

• Use the inverse operations in reverse order of appearance. Start from the outside and work inward.

• Always verify. A quick plug-in check can save you from second-guessing later.

Final thought: math is a conversation with structure

Problems like 3(x - 4) = 12 aren’t just about getting a number for x. They’re about listening to the structure of the equation—recognizing the multiplication, spotting the bracket, and using the right inverse move to reveal the hidden variable. With that mindset, you’ll find yourself approaching similar questions with greater ease, clarity, and a touch more confidence.

If you want a couple more challenges to test this approach, you can try the quick variations above or craft a few on your own. The pattern is the same, and that’s what makes it feel almost effortless after you’ve seen it a few times.