Hey math enthusiasts! Today, we're diving into the world of inequalities, specifically tackling the problem: 9^(3-x) ≥ (1/81)^(2x-5). Don't worry if it looks a bit intimidating at first; we'll break it down step by step to make sure you totally get it. We'll explore the core concepts, the necessary steps to solve it, and how to arrive at the solution. Let's start with the basics.

Demystifying the Inequality

First things first, let's understand what an inequality actually is. Unlike equations, which use an equals sign (=), inequalities use symbols like greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤). These symbols indicate a range of values, not just a single point. In our case, the inequality 9^(3-x) ≥ (1/81)^(2x-5) asks us to find all the values of x for which 9 raised to the power of (3-x) is greater than or equal to (1/81) raised to the power of (2x-5). This is super important because it's not just one specific number, but a range of numbers that satisfy this condition.

Before we jump into the steps, remember that the goal is to find the values of x that make the left side of the inequality larger than or equal to the right side. This means we're looking for all the numbers that, when plugged into the equation, would make the statement true. Think of it like this: imagine a seesaw. The inequality is telling us that one side of the seesaw (the left side, 9^(3-x)) must be either higher than or at the same level as the other side (the right side, (1/81)^(2x-5)). Our job is to figure out what values of x allow the seesaw to balance or tilt in favor of the left side. To make things a little easier to understand, let's consider some key concepts before we get into the solving part. We’ll be using the properties of exponents. Exponents tell us how many times a number (the base) is multiplied by itself. Also, we will use the concept of converting numbers to the same base. This will simplify the problem making it easier to solve. We’re working with exponential functions, which means our variables are in the exponent. Cool, right? So, by changing the base to be the same on both sides of the inequality, we make things more manageable. We're getting closer to understanding the range of x that satisfies this condition.

Step-by-Step Solution

Alright, let's get down to the nitty-gritty of solving this inequality! Here's a breakdown of the steps:

Step 1: Matching the Bases

The first crucial step is to get the bases of both sides of the inequality to be the same. This allows us to compare the exponents directly. Notice that both 9 and 81 are powers of 3. We can rewrite 9 as 3^2 and 81 as 3^4. Also, keep in mind that 1/81 can be expressed as 81^-1. Thus, we can rewrite the initial inequality as:

(3²⁾(3-x) ≥ (3^-4)(2x-5)

By expressing both sides of the inequality with the same base, we've set the stage for comparing the exponents directly. This simplifies the equation significantly. Remember that the ultimate goal is to find a single, manageable expression to work with. Matching the bases is the first giant leap towards making that happen. Now, let’s move to the next part.

Step 2: Simplifying the Exponents

With the bases matching, we can now simplify the exponents using the power of a power rule: (a^m)n = a^(m*n). Applying this rule to both sides, we get:

3^(2(3-x)) ≥ 3^(-4(2x-5))**

Which simplifies to:

3^(6-2x) ≥ 3^(-8x+20)

Now, because the bases are the same (both are 3), we can drop the bases and compare the exponents directly. This leads us to a simpler inequality that is much easier to solve. We're cutting through the layers of complexity and zeroing in on a solution that is within reach.

Step 3: Comparing the Exponents

Since the bases are identical, the inequality holds true if and only if the exponent on the left side is greater than or equal to the exponent on the right side. So, we're now comparing: 6 - 2x ≥ -8x + 20. This is way easier to deal with, right? Now, let's solve this linear inequality to find our range for x. This simplification is a testament to the power of matching bases and simplifying exponents. It's like we're stripping away all the extra fluff and getting right to the core of the problem.

Step 4: Solving for x

Let's isolate x. Add 8x to both sides: 6 + 6x ≥ 20. Subtract 6 from both sides: 6x ≥ 14. Divide both sides by 6: x ≥ 14/6. Simplify the fraction: x ≥ 7/3. So, the solution is x ≥ 7/3. This is it! We have found the range of values for x that satisfies our initial inequality. Any value of x that is greater than or equal to 7/3 will make the original statement true. This means, if we plug any number greater than or equal to 7/3 into the original expression (9^(3-x) ≥ (1/81)^(2x-5)), the inequality holds.

Conclusion: The Final Answer

Therefore, the solution to the inequality 9^(3-x) ≥ (1/81)^(2x-5) is x ≥ 7/3. This means that any value of x that is greater than or equal to 7/3 will satisfy the original inequality.

This is a journey. We started with what seemed like a complicated exponential inequality and, step by step, broke it down into something manageable. We used the properties of exponents, changed the base to be the same, simplified, and then, with some basic algebra, found the range of values for x that worked. Math might seem daunting sometimes, but with a methodical approach, it becomes very accessible. So, go ahead and practice, try different problems, and you'll find yourself getting better at it. You got this, guys! Keep up the great work, and happy solving!

Visualizing the Solution

To really cement your understanding, it helps to visualize the solution on a number line. On a number line, we'll mark 7/3 (which is approximately 2.33). Because the solution includes 'greater than or equal to', we use a closed circle (filled-in) at 7/3. Then, we shade the number line to the right of 7/3, because all values greater than 7/3 are part of our solution. This visual representation clarifies that the solution includes 7/3 and all numbers larger than it. You can see how the solution spans to infinity on the number line, confirming that there are infinite solutions that will satisfy the inequality.

Key Takeaways and Tips

Let's recap what we've learned and add some tips to your math toolkit:

• Matching Bases is Key: This is the most crucial step when solving exponential inequalities. Always look for a common base. This simplifies the problem significantly.
• Exponent Rules: Brush up on your exponent rules! Knowing the rules for powers of powers, multiplication, and division of exponents is essential.
• Inequality Signs: Remember that when multiplying or dividing both sides of an inequality by a negative number, you must flip the inequality sign. Luckily, we didn't have to do this in our example, but it's a common trick to watch out for.
• Practice: The more problems you solve, the more comfortable you will become. Try different variations of exponential inequalities to solidify your understanding.
• Check Your Work: Always plug your solution (or a value within your solution range) back into the original inequality to make sure it works. This helps you catch any mistakes you might have made along the way.

By following these steps and tips, you'll be well-equipped to tackle similar problems in the future. Keep practicing, and you'll find that solving inequalities becomes easier with each try! Keep the momentum going. You're doing great.