Solving The Inequality 4(3x-1) < -2(3x+2) A Step-by-Step Guide

Jul 15, 2025 by qnaunigon 63 views

Unraveling the Inequality: A Step-by-Step Solution to 4(3x-1) < -2(3x+2)

Introduction

Hey guys! Today, we're diving into the world of inequalities with a classic problem that many students encounter in algebra. We're going to break down the inequality 4(3x - 1) < -2(3x + 2) step-by-step, so you can not only understand the solution but also master the process for tackling similar problems. Inequalities might seem a bit daunting at first, but with a clear approach and a bit of practice, you'll be solving them like a pro in no time. So, let's jump right in and unravel this mathematical puzzle together!

This article aims to provide a comprehensive guide to solving this inequality, making it super easy for everyone to understand. Whether you're a student brushing up on your algebra skills or just someone curious about math, we've got you covered. We'll start by explaining the basic concepts and then move on to the step-by-step solution. By the end of this article, you'll not only know the answer but also understand the why behind each step. Trust me, with a solid understanding of the fundamentals, inequalities become much less intimidating and even, dare I say, fun!

Breaking Down the Problem: 4(3x - 1) < -2(3x + 2)

Our mission, should we choose to accept it (and we do!), is to find the values of 'x' that make the inequality 4(3x - 1) < -2(3x + 2) true. This essentially means we need to isolate 'x' on one side of the inequality. To do this, we'll use the same principles we use for solving equations, with one important twist: we need to be mindful of how multiplying or dividing by a negative number affects the inequality sign. But don't worry, we'll get to that in due time. For now, let's focus on simplifying the inequality by getting rid of those parentheses.

The first thing we need to do is tackle those parentheses. Remember the distributive property? It's our best friend here. We're going to multiply the numbers outside the parentheses by each term inside. So, on the left side, we have 4 multiplied by both 3x and -1. On the right side, we have -2 multiplied by both 3x and +2. Let's break it down:

4 * (3x) = 12x
4 * (-1) = -4
-2 * (3x) = -6x
-2 * (+2) = -4

Now, we can rewrite our inequality, replacing the expressions in parentheses with their expanded forms. This gives us a new, simplified version of the inequality that's much easier to work with. We're essentially transforming the problem into a more manageable form, one step at a time. This is a key strategy in mathematics – breaking down complex problems into simpler, more digestible pieces. So, let's see what our inequality looks like now.

Step-by-Step Solution

Now that we've expanded both sides of the inequality, we have a clearer picture of what we're working with. Our inequality now looks like this: 12x - 4 < -6x - 4. This is a significant step forward because we've eliminated the parentheses, making the next steps much easier to execute. The goal remains the same: to isolate 'x' on one side of the inequality. To do this, we'll need to gather all the 'x' terms on one side and all the constant terms on the other. This is similar to solving equations, and we'll use the same principles of adding or subtracting terms from both sides to maintain the balance of the inequality.

Combining Like Terms

The next step is to gather all the 'x' terms on one side of the inequality and the constant terms on the other. To do this, we'll perform operations on both sides of the inequality, ensuring that we maintain the balance. Let's start by moving the '-6x' term from the right side to the left side. To do this, we'll add '6x' to both sides of the inequality. This is a crucial step, as it allows us to consolidate the 'x' terms and move closer to isolating 'x'.

Adding 6x to both sides, we get:

12x - 4 + 6x < -6x - 4 + 6x

Simplifying this, we combine the 'x' terms on the left side (-6x + 6x = 0 on the right side), resulting in:

18x - 4 < -4

Now, we need to move the constant term '-4' from the left side to the right side. To do this, we'll add '4' to both sides of the inequality. This will cancel out the '-4' on the left side, leaving us with just the 'x' term and its coefficient. Remember, whatever we do to one side of the inequality, we must do to the other to keep the inequality balanced.

Adding 4 to both sides, we have:

18x - 4 + 4 < -4 + 4

Simplifying this gives us:

18x < 0

We're getting closer! We've successfully isolated the 'x' term on the left side, but it's still attached to the coefficient '18'. Our next step is to get rid of this coefficient so that we have 'x' all by itself. This will give us the solution to our inequality, telling us the range of values for 'x' that make the inequality true.

Isolating x

We're now at the final stage of isolating 'x'. We have the inequality 18x < 0. To get 'x' by itself, we need to get rid of the coefficient '18'. Since '18' is multiplying 'x', we'll do the opposite operation: we'll divide both sides of the inequality by '18'. This is a standard algebraic technique, and it's crucial for solving both equations and inequalities.

Dividing both sides by 18, we get:

(18x) / 18 < 0 / 18

Simplifying this, we find:

x < 0

And there you have it! We've successfully isolated 'x' and found the solution to our inequality. The solution x < 0 tells us that any value of 'x' that is less than 0 will make the original inequality true. This is a concise and powerful statement, and it's the culmination of all our hard work. But we're not quite done yet. It's always a good idea to double-check our work and make sure our solution makes sense. Let's take a moment to verify our answer and ensure we haven't made any mistakes along the way.

Verifying the Solution

Alright, we've arrived at our solution: x < 0. But before we celebrate, let's take a moment to verify our answer. This is a crucial step in problem-solving, as it helps us catch any potential errors and ensures that our solution is correct. Verifying our solution is like a mini-audit, giving us the confidence that we've done everything right. There are a couple of ways we can do this. One way is to substitute a value for 'x' that satisfies our solution (i.e., a number less than 0) into the original inequality and see if it holds true. Another way is to rethink our steps and make sure each one was logically sound.

Let's take the first approach and choose a value for 'x' that is less than 0. A simple choice would be x = -1. Now, we'll substitute this value into our original inequality: 4(3x - 1) < -2(3x + 2).

Substituting x = -1, we get:

4(3(-1) - 1) < -2(3(-1) + 2)

Now, let's simplify both sides of the inequality:

4(-3 - 1) < -2(-3 + 2)

4(-4) < -2(-1)

-16 < 2

Is this true? Yes, -16 is indeed less than 2. This confirms that our solution x < 0 is likely correct. If we had obtained a false statement (e.g., -16 > 2), it would indicate that we made a mistake somewhere along the way, and we would need to go back and re-examine our steps. But since our test case worked out, we can be reasonably confident in our solution. However, it's always a good idea to try another test case just to be extra sure. Let's try x = -2.

Substituting x = -2 into the original inequality:

4(3(-2) - 1) < -2(3(-2) + 2)

4(-6 - 1) < -2(-6 + 2)

4(-7) < -2(-4)

-28 < 8

Again, this is a true statement, further reinforcing our confidence in our solution. We've successfully verified our solution using two different test cases, which is a strong indication that we've solved the inequality correctly. Now that we've confirmed our answer, we can confidently move on to the next problem, knowing that we've mastered this particular type of inequality.

Conclusion

Wow, guys, we did it! We successfully solved the inequality 4(3x - 1) < -2(3x + 2) and found that the solution is x < 0. We didn't just get the answer, though. We also walked through the entire process step-by-step, explaining the why behind each move. We started by breaking down the problem, using the distributive property to eliminate parentheses. Then, we combined like terms, carefully moving 'x' terms to one side and constants to the other. Finally, we isolated 'x' by dividing both sides of the inequality by its coefficient. And, perhaps most importantly, we verified our solution to make sure we got it right. This is the kind of thorough approach that will serve you well in all your mathematical endeavors.

Understanding inequalities is a fundamental skill in algebra and beyond. It's not just about finding the right answer; it's about developing a logical and systematic approach to problem-solving. The steps we followed today – expanding, simplifying, isolating the variable, and verifying – are applicable to a wide range of mathematical problems. So, the next time you encounter an inequality, remember the strategies we've discussed here. Break the problem down, take it one step at a time, and don't forget to verify your solution. You've got this!

So, what's the big takeaway here? Well, it's that inequalities, like many things in math, become much less scary when you approach them with a clear plan and a solid understanding of the underlying principles. We've shown you how to tackle this particular inequality, but the real victory is in the skills you've gained along the way. You now have a powerful set of tools for solving similar problems, and that's something to be proud of. Keep practicing, keep exploring, and keep pushing your mathematical boundaries. The world of math is vast and fascinating, and there's always something new to discover. And remember, we're here to help you on your journey, one step at a time.