Solving The Equation (3x + 2) / 4 - 2 = 3

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Hey everyone! Let's dive into solving the equation (3x + 2) / 4 - 2 = 3. Don't worry, it might look a little intimidating at first, but we'll break it down step by step and make it super easy to understand. This is a classic algebra problem, and mastering it will give you a solid foundation for more complex equations. Ready to get started? Let's go!

Understanding the Basics: Equations and Variables

Alright, before we jump into the nitty-gritty, let's refresh our memory on what an equation and a variable actually are. An equation, in simple terms, is a mathematical statement that shows two expressions are equal. It's like a balanced scale, where both sides must have the same value. The equals sign (=) is the key symbol here; it tells us that everything on the left side is the same as everything on the right side. Got it?

Now, let's talk about variables. A variable is a letter or symbol (like 'x' in our equation) that represents an unknown number. Our goal in solving an equation is to find the value of this unknown variable. Think of it like a puzzle where we're trying to figure out what number 'x' stands for, so we can make both sides of the equation balance out perfectly. When we solve an equation, we're essentially isolating the variable (getting it by itself) on one side of the equation.

So, our mission is to figure out what number 'x' needs to be, so that when we do the math, the left side of the equation becomes the same as the right side, which is 3. That's the essence of solving equations! We want to find the value of the unknown that makes the equation true. Knowing these basic concepts will make solving these types of problems a breeze. Remember that it's all about keeping both sides balanced. Any operation performed on one side MUST be performed on the other to maintain equality. Let's get our hands dirty!

Step-by-Step Solution: Unraveling the Equation

Okay, guys, let's roll up our sleeves and tackle this equation step by step. We'll be using inverse operations to isolate the variable 'x'. Inverse operations are basically the opposite actions: addition and subtraction are inverses, as are multiplication and division. Remember, our main goal is to get 'x' all by itself on one side of the equals sign. Let's do this!

  1. Isolate the Term with 'x': The first thing we need to do is get rid of that pesky '-2' on the left side. To do that, we perform the inverse operation: addition. We add 2 to both sides of the equation. This maintains the balance:

    (3x + 2) / 4 - 2 + 2 = 3 + 2

    This simplifies to:

    (3x + 2) / 4 = 5

  2. Eliminate the Division: Now, we have a division by 4. To get rid of that, we do the inverse operation, which is multiplication. We multiply both sides of the equation by 4:

    [(3x + 2) / 4] * 4 = 5 * 4

    This simplifies to:

    3x + 2 = 20

  3. Isolate the Term with 'x' Again: Next, we want to get the term with 'x' alone. We need to get rid of the '+2'. So, we perform the inverse operation and subtract 2 from both sides:

    3x + 2 - 2 = 20 - 2

    This simplifies to:

    3x = 18

  4. Solve for 'x': Finally, we have '3x = 18'. To solve for 'x', we need to divide both sides by 3:

    3x / 3 = 18 / 3

    This simplifies to:

    x = 6

And there you have it, folks! We've found that x = 6. We've gone from a somewhat complex-looking equation to a simple solution through a series of logical steps. Congratulations, you are now one step closer to mastering algebra.

Verification: Checking Your Answer

Awesome, now that we've found our answer (x = 6), let's check it! It's always a good idea to verify our solution to make sure we didn't make any mistakes along the way. Verification is like a built-in safety net, helping us catch errors and build confidence in our problem-solving skills. So how do we do it? We substitute the value of 'x' we found back into the original equation and see if both sides equal each other. If they do, then our solution is correct! Let’s do it!

Our original equation was: (3x + 2) / 4 - 2 = 3

  1. Substitute x = 6: Replace 'x' with 6 in the equation:

    (3 * 6 + 2) / 4 - 2 = 3

  2. Simplify: Now, let's simplify the equation step by step:

    (18 + 2) / 4 - 2 = 3

    20 / 4 - 2 = 3

    5 - 2 = 3

    3 = 3

And there you have it! Since both sides of the equation are equal, our answer, x = 6, is correct. Nice work, everyone! Verification is a vital step in algebra; it confirms the accuracy of our solutions, strengthens our problem-solving skills, and helps us build confidence. It’s a good habit to get into. Just remember to always plug your answer back into the original equation, and if both sides match, you've got it!

Key Takeaways and Tips for Success

Alright, guys, let's wrap things up by highlighting some key takeaways and providing tips to help you become algebra pros. By now, you've successfully navigated the equation (3x + 2) / 4 - 2 = 3, and you've seen how a systematic approach and careful attention to detail can transform a seemingly complex problem into a manageable one. Here's a quick recap of what we've learned and some helpful tips to keep you on the right track.

  1. Inverse Operations: Remember that inverse operations are your best friends. Use addition to undo subtraction, subtraction to undo addition, multiplication to undo division, and division to undo multiplication. This is the cornerstone of solving equations, so make sure you understand this concept well.
  2. Balance is Key: Always remember to perform the same operation on both sides of the equation. This ensures that the equation remains balanced and that your solution is valid. Think of it like a scale; you have to do the same thing on both sides to keep things even.
  3. Step-by-Step Approach: Break down complex problems into smaller, more manageable steps. This will help you avoid making mistakes and keep track of your progress. Work systematically, one step at a time, and you'll find the process much less daunting.
  4. Practice Makes Perfect: The more you practice, the better you'll become at solving equations. Work through various problems, starting with simpler ones and gradually increasing the difficulty. This will build your confidence and proficiency.
  5. Check Your Work: Always verify your solution by substituting the value of the variable back into the original equation. This is a crucial step to ensure the accuracy of your answer and catch any errors you may have made.
  6. Seek Help When Needed: Don't hesitate to ask for help if you're struggling. Whether it's your teacher, a classmate, or an online resource, getting clarification can save you a lot of time and frustration.

By following these tips and practicing regularly, you'll be well on your way to becoming an algebra whiz! Remember that patience and persistence are key, and don’t be afraid to make mistakes. Mistakes are a natural part of the learning process; they help you identify areas where you need to improve and reinforce your understanding. Keep at it, and you'll see your skills grow and your confidence soar. You've got this, everyone!