Hey guys! Let's dive into solving inequalities. Specifically, we're going to break down how to tackle the inequality: 9^(3x - 2) < 1 / 81^(2x - 5). This might look a little intimidating at first, but trust me, it's totally manageable once you break it down into steps. We will go through this step by step, so even if you're new to this, you will be able to follow along. So, grab your pencils and let's get started. The key to solving these types of inequalities lies in understanding exponential rules and how to manipulate exponents. Remember that inequalities follow similar rules to equations, but with a crucial twist: when you multiply or divide by a negative number, you must flip the inequality sign. Keep this in mind, and you will be able to easily solve this kind of inequalities. Before we jump into solving the specific inequality, let's brush up on some essential concepts. This will help make sure we're all on the same page. So that we can ensure that everyone understands the concepts, let's explore these concepts first! We will explore the exponential rules. These are going to be useful as we solve this inequality, so make sure you pay close attention!

Understanding the Basics

First things first, let's talk about the fundamentals. Inequalities are mathematical statements that compare two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Unlike equations, which state that two expressions are equal, inequalities describe a range of values. The goal when solving an inequality is to find the set of values for the variable that makes the inequality true. This is often represented on a number line or in interval notation. Now, let's get down to the important bit. We need to remember a few key properties of exponents before we start: a^(m) * a^(n) = a^(m+n) and (a^(m))(n) = a^(m*n). Make sure you understand these rules since they are going to be the basis of solving the inequality. These rules let us simplify and manipulate exponential expressions, which is exactly what we need to do to solve our inequality. Remember also that, when possible, it's super helpful to express both sides of the inequality with the same base. This will allow us to easily compare the exponents. So in order to ensure that we will solve it correctly, let's go over how we can use these rules to simplify.

Simplifying the Inequality

Okay, here's where the rubber meets the road. Our inequality is: 9^(3x - 2) < 1 / 81^(2x - 5). The first step is to express both sides of the inequality with the same base. Notice that both 9 and 81 are powers of 3 (9 = 3^2, and 81 = 3^4). This means we can rewrite the inequality so that it has the same base. Let's start with rewriting 9 as 3^2 and 81 as 3^4. So our inequality becomes (3²⁾(3x - 2) < 1 / (3⁴⁾(2x - 5). Now, use the power of a power rule: (a^m)n = a^(m*n). This simplifies to 3^(6x - 4) < 1 / 3^(8x - 20). Next, we need to bring that 3^(8x - 20) in the denominator to the numerator. Remember that a^(-n) = 1 / a^(n). So, we can rewrite 1 / 3^(8x - 20) as 3^-(8x - 20), which simplifies to 3^(-8x + 20). At this point, our inequality is: 3^(6x - 4) < 3^(-8x + 20). Now that we have the same base on both sides, we can focus on comparing the exponents. This is the whole reason we went through all the trouble of changing the bases. It lets us cut to the chase and directly compare the powers. Pretty neat, right? Now let us make sure we understand the concepts before we proceed to the next step.

Comparing the Exponents

Since the bases are the same (both are 3), we can now compare the exponents directly. If 3^(6x - 4) < 3^(-8x + 20), then 6x - 4 < -8x + 20. This is the core step! By ditching the exponential form and just focusing on the exponents, we've transformed the problem into a simple linear inequality that we know how to solve. Now, let's solve for x. First, we'll add 8x to both sides: 6x - 4 + 8x < -8x + 20 + 8x. This gives us 14x - 4 < 20. Next, add 4 to both sides: 14x - 4 + 4 < 20 + 4. This simplifies to 14x < 24. Finally, we divide both sides by 14: 14x / 14 < 24 / 14. This gives us x < 24/14, which can be simplified to x < 12/7. So, the solution to the inequality is x < 12/7. This means any value of x less than 12/7 will make the original inequality true. We can express this solution in a few different ways: using inequality notation (x < 12/7), on a number line (with an open circle at 12/7 and an arrow pointing to the left), or in interval notation (-∞, 12/7). These are all just different ways of saying the same thing.

Final Answer and Conclusion

Alright, folks, we've done it! We've successfully solved the inequality 9^(3x - 2) < 1 / 81^(2x - 5). The final answer is x < 12/7. Remember that solving inequalities requires a solid understanding of exponential rules, especially the power of a power rule and the ability to work with fractions. Expressing numbers with the same base is the key trick to simplifying and solving this kind of problem. Make sure to double-check your work, particularly when dealing with negative numbers or when flipping the inequality sign. Always check your answer to make sure it makes sense in the context of the original inequality. You can do this by plugging in a value less than 12/7 and checking if it satisfies the inequality. If it does, great! You know you're on the right track. If not, go back and review your steps. If you have any questions, don't hesitate to ask. Practice makes perfect, so keep working through problems. You'll get better and better with each one. Keep in mind that with practice, you'll become more comfortable and confident in solving these types of problems. Remember to always double-check your work and to express your answers clearly using the different notations. That's all for today, guys! Keep practicing, and you'll be inequality-solving pros in no time. Congratulations on making it through this guide, you now know how to solve the inequality! Keep up the good work and keep learning!