Hey everyone! Today, we're diving into the world of exponential inequalities. Specifically, we're going to break down how to solve an inequality like this: 9^(3-x) ≥ 1/(81^(2x-5)). Don't worry, it might look a little intimidating at first, but trust me, it's totally manageable. We'll walk through it step-by-step, making sure you understand every move. Ready? Let's get started!

Understanding the Basics: Exponential Inequalities

First things first, what exactly is an exponential inequality? Well, it's an inequality where the variable (that's the 'x' we're trying to find) is in the exponent. Think of it as an equation, but instead of an equals sign (=), we have an inequality sign like greater than (>) or less than (<). Solving these kinds of problems involves a few key principles. The goal is to isolate the variable. We'll be using our knowledge of exponents and logarithms to simplify things and get to the solution. This is all about finding the range of values for 'x' that make the inequality true. The core idea is to manipulate the expressions until we have the same base on both sides of the inequality. From there, we can compare the exponents directly. Remember, the base is the number being raised to a power. When the bases are the same, we can focus on comparing the exponents. This is the main strategy we'll employ. Another important point is to remember how to deal with fractions and negative exponents. We'll be sure to cover these concepts thoroughly as we work through the problem. This is a crucial step towards understanding the concepts. It builds the foundation for solving more complex inequality problems in the future. Understanding this is key to solving a vast range of problems. So, buckle up; it's going to be a fun ride!

Step-by-Step Solution: Breaking Down the Inequality

Alright, let's get down to business and solve that inequality: 9^(3-x) ≥ 1/(81^(2x-5)). We'll go through this systematically. The first thing we want to do is to rewrite everything with the same base. You'll notice that both 9 and 81 can be expressed as powers of 3. This is our golden ticket! Since 9 is 3 squared (3^2) and 81 is 3 to the power of 4 (3^4), we can rewrite our inequality. This step is all about making things simpler and easier to compare. Remember, when you have a power raised to another power, you multiply the exponents. Let's start with the left side. We have 9^(3-x), which we can rewrite as (3²⁾(3-x). Using the power of a power rule, this becomes 3^(2*(3-x)) or 3^(6-2x). On the right side, we have 1/(81^(2x-5)). We know 81 is 3^4, so we can rewrite this as 1/((3⁴⁾(2x-5)) which simplifies to 1/(3^(8x-20)). Now, we can use the property that 1/a^b = a^(-b) to further simplify it to 3^-(8x-20) or 3^(-8x+20). Now our inequality is 3^(6-2x) ≥ 3^(-8x+20). Since both sides have the same base (3), we can compare the exponents. That means we now consider the exponents: 6-2x ≥ -8x+20. Isn't that neat? By changing the expressions to having a common base, we are able to compare the exponents directly. From this point, we just have to solve a simple linear inequality. This is the core strategy, converting the original inequality into a form that's much easier to work with. Remember, the trick is to get those bases to match! By knowing your rules of exponents, you can simplify the problem significantly. This entire process becomes easier with practice.

Isolating the Variable: Solving for 'x'

Now that we've got the same base on both sides, it's time to solve for 'x'. We have the inequality: 6 - 2x ≥ -8x + 20. The first step is to combine the 'x' terms. Let's add 8x to both sides. This gives us 6 + 6x ≥ 20. Cool, right? Now we only have the 'x' terms on one side. Next, we need to get rid of that pesky 6. We can subtract 6 from both sides of the inequality. This simplifies to 6x ≥ 14. Getting closer! Our last step is to isolate 'x'. We can divide both sides by 6. This gives us x ≥ 14/6. Then we can simplify the fraction 14/6 to 7/3. So, the solution to our inequality is x ≥ 7/3. We have now successfully isolated the variable! This means any value of 'x' that is greater than or equal to 7/3 will satisfy the original inequality. Pretty simple, once you get the hang of it, right? This step is a straightforward application of algebraic manipulation. We're just applying the same operations to both sides to maintain the balance and solve for 'x'. These steps are the building blocks of solving complex problems. Remember to always check your answer! You can plug a value greater than or equal to 7/3 back into the original inequality to make sure it holds true.

Checking Your Work: Verifying the Solution

It's always a good idea to check your solution. Let's pick a value for 'x' that is greater than or equal to 7/3. A nice easy number would be 3. If x = 3, then the original inequality is 9^(3-3) ≥ 1/(81^(23-5)). This simplifies to 9^0 ≥ 1/(81^1), which means 1 ≥ 1/81. This is true! Which means our solution, x ≥ 7/3, appears to be correct. Let's also try plugging in 7/3 to make sure the equality holds. 9^(3-7/3) ≥ 1/(81^(2(7/3)-5)). This simplifies to 9^(2/3) ≥ 1/(81^(-1/3)). Rewriting, we get (3²⁾(2/3) ≥ (3⁴⁾(-1/3) which becomes 3^(4/3) ≥ 3^(-4/3). This is true. So now, we've confirmed the solution. This is a very important step! It ensures we haven't made any mistakes during our calculations. Always double-check your work to avoid any potential errors. Verification is also a great way to reinforce the concepts and improve your understanding. Taking the time to do this builds confidence in your answers. It also makes sure you are right before you submit your solution. Doing a quick check helps you identify any potential errors before you consider the problem complete.

Conclusion: Mastering Exponential Inequalities

And there you have it! We've successfully solved the exponential inequality: 9^(3-x) ≥ 1/(81^(2x-5)). We started by understanding the basics of exponential inequalities, simplifying using the same base, isolating the variable, and then verifying the solution. The key takeaway here is to remember the rules of exponents and how to manipulate equations to get a common base. Once you've done that, you can compare the exponents and solve for the variable. Keep practicing, and you'll become a pro at these problems in no time. Congratulations, you've conquered another math challenge! Remember, practice makes perfect. The more you work with exponential inequalities, the easier they'll become. So, keep up the great work, and don't be afraid to tackle new challenges. By breaking down complex problems into smaller, manageable steps, anyone can learn to solve them. Keep practicing, and you'll find these types of problems much easier. You're building a strong foundation in algebra. Keep learning and have fun! You've got this!