Missing Step Solving 5-8x < 2x + 3 Inequality

Hey guys! Let's dive into the world of inequalities and solve a puzzle together. We're tackling the inequality 5 - 8x < 2x + 3, and our mission is to find the missing step in its solution. It's like being a math detective, piecing together clues to crack the case! So, buckle up, sharpen your pencils (or keyboards!), and let's get started.

Step 1 The Foundation of Our Inequality Journey

Our journey begins with the inequality 5 - 8x < 2x + 3. The first step, as our problem states, is to subtract 3 from both sides. This is a fundamental move in solving inequalities (and equations, for that matter). Why do we do this? Well, our ultimate goal is to isolate 'x' on one side of the inequality. Subtracting 3 helps us move closer to that goal by simplifying the inequality. When we subtract 3 from both sides, we maintain the balance of the inequality, ensuring that our solution remains valid.

So, let’s do it: 5 - 8x - 3 < 2x + 3 - 3. This simplifies to 2 - 8x < 2x. Notice how we've eliminated the constant term on the right side, making the inequality a bit cleaner. This is a crucial step because it sets the stage for the next operation, which will bring us even closer to isolating 'x'. Think of it as laying the groundwork for the rest of our solution. Without this step, we'd be trying to build on an uneven foundation, making the process much more difficult. In the world of math, just like in real life, a solid foundation is key to success!

Step 2 The Missing Link Bringing 'x' Terms Together

Now, we arrive at the heart of our puzzle: the missing step. After subtracting 3 from both sides, we're left with 2 - 8x < 2x. Looking at this inequality, what's the most logical next move? Remember, our goal is to isolate 'x'. Currently, we have 'x' terms on both sides of the inequality. To simplify things, we need to gather these terms together. This is where the missing step comes into play, and it's a critical one for solving this inequality effectively.

Considering our options, the most strategic move here is to add 8x to both sides of the inequality. Why? Because this will eliminate the '-8x' term on the left side, effectively moving all the 'x' terms to the right side. This is a classic algebraic technique – moving variables to one side and constants to the other. It's like sorting your toys or organizing your closet; grouping similar items together makes everything easier to manage. So, let's add 8x to both sides: 2 - 8x + 8x < 2x + 8x. This simplifies to 2 < 10x. See how much cleaner that looks? We've successfully moved all the 'x' terms to one side, and we're one step closer to isolating 'x' completely. This step is the bridge that connects the initial simplification to the final isolation of our variable.

Step 3 The Grand Finale Isolating 'x' to Solve the Inequality

We've reached the final act of our inequality drama! We're at 2 < 10x, and the finish line – isolating 'x' – is in sight. To get 'x' all by itself, we need to deal with the coefficient 10. Remember, the coefficient is the number multiplied by the variable. In this case, 'x' is being multiplied by 10. To undo this multiplication, we need to perform the inverse operation: division.

So, our final step is to divide both sides of the inequality by 10. This is a fundamental rule in algebra: whatever you do to one side of an equation or inequality, you must do to the other to maintain balance. Dividing both sides by 10 gives us 2/10 < 10x/10, which simplifies to 1/5 < x or x > 1/5. We've done it! We've successfully isolated 'x' and solved the inequality. This means that any value of 'x' greater than 1/5 will satisfy the original inequality 5 - 8x < 2x + 3. This final division is the key that unlocks the solution, revealing the range of values for 'x' that make the inequality true. It's the culmination of all our previous steps, bringing us to the satisfying conclusion of our mathematical journey.

Choosing the Correct Path Option A Detailed Analysis

Let's circle back to our original question: What is the missing step? We were given two options:

A. Add 2x to both sides of the inequality.

After our detailed walkthrough, it's clear that this is indeed the missing step. Adding 2x to both sides after subtracting 3 was the crucial move that allowed us to gather the 'x' terms on one side, paving the way for isolating 'x'. This step is not just a random operation; it's a strategic maneuver that follows the fundamental principles of algebra. It demonstrates our understanding of how to manipulate inequalities while maintaining their balance and integrity.

If we hadn't added 2x, we would have been stuck with 'x' terms on both sides, making it impossible to isolate 'x' and solve the inequality. This highlights the importance of choosing the correct operation at each step. Solving inequalities (and equations) is like following a recipe; each step must be performed in the correct order to achieve the desired outcome. Adding 2x at this stage was the perfect ingredient to move our solution forward, bringing us closer to the final answer. It's a testament to the power of strategic thinking in mathematics, where every move counts.

Why Other Options Don't Fit Understanding the Flow of the Solution

Now, let's briefly consider why other options might not fit as the missing step. This is crucial for developing a deeper understanding of the problem-solving process. Often, understanding why a particular step is correct also involves understanding why other steps are incorrect or less efficient. In our case, we had one primary option provided, but in more complex scenarios, there might be multiple plausible-sounding steps. Evaluating these alternatives helps us hone our mathematical intuition and decision-making skills.

For instance, we might consider other operations like adding or subtracting constants, but these wouldn't directly address our primary goal of gathering 'x' terms. Similarly, multiplying or dividing by a constant at this stage might complicate the inequality without bringing us closer to isolating 'x'. The key is to choose the operation that most efficiently moves us towards our goal. In this case, adding 2x was the most direct and effective way to simplify the inequality and set the stage for the final step of dividing by the coefficient of 'x'. By understanding the flow of the solution and the purpose of each step, we can confidently navigate through mathematical problems and arrive at the correct answer. It's like learning a dance; once you understand the sequence of steps, you can execute the routine with grace and precision.

Mastering Inequalities Key Takeaways for Mathematical Success

We've successfully navigated the missing step in solving the inequality 5 - 8x < 2x + 3! But the journey doesn't end here. Understanding the underlying principles is key to mastering inequalities and other mathematical concepts. So, let's recap some key takeaways that will help you tackle similar problems with confidence.

First, remember the importance of isolating the variable. This is the ultimate goal in solving any equation or inequality. It's like finding the treasure at the end of a map; all our steps are directed towards this final destination. Second, understand the power of inverse operations. Addition and subtraction are inverses, as are multiplication and division. Using these operations strategically allows us to undo operations and simplify expressions. It's like having a set of magical tools that can transform equations and inequalities into simpler forms. Third, always maintain balance. Whatever operation you perform on one side of an equation or inequality, you must perform on the other side. This ensures that the solution remains valid. It's like walking a tightrope; you need to maintain your balance to reach the other side safely.

Finally, practice makes perfect. The more you work with inequalities, the more comfortable and confident you'll become. It's like learning a new language; the more you practice, the more fluent you'll become. So, keep exploring, keep questioning, and keep solving! The world of mathematics is full of fascinating puzzles, and with the right tools and mindset, you can unlock them all. Remember, every problem is an opportunity to learn and grow. So, embrace the challenge, and let's continue our mathematical adventures together!

Conclusion The Beauty of Mathematical Problem-Solving

In conclusion, we've not only found the missing step in solving the inequality 5 - 8x < 2x + 3, but we've also delved into the underlying principles that make this solution possible. We've seen how subtracting constants, adding variable terms, and dividing by coefficients all play crucial roles in isolating the variable and revealing the solution. This journey highlights the beauty of mathematical problem-solving – the logical progression, the strategic decision-making, and the satisfying feeling of arriving at the correct answer.

Solving inequalities is more than just manipulating numbers and symbols; it's about developing critical thinking skills, problem-solving strategies, and a deep understanding of mathematical concepts. It's like learning to play a musical instrument; it requires practice, patience, and a willingness to learn from mistakes. But the rewards are immense – the ability to analyze complex situations, make informed decisions, and appreciate the elegance and power of mathematics.

So, the next time you encounter an inequality or any mathematical challenge, remember the lessons we've learned today. Embrace the process, break the problem down into smaller steps, and trust in your ability to find the solution. And who knows, you might even discover a missing step or two along the way! Keep exploring the fascinating world of mathematics, and let's continue to unravel its mysteries together.