Hey guys! Today, we're diving into the exciting world of exponential inequalities. Specifically, we're going to tackle the inequality: 9^(3x+2) < 1/81^(2x+5). Don't worry if it looks intimidating at first. We'll break it down step-by-step, so you'll be solving these like a pro in no time!

Understanding Exponential Inequalities

Before we jump into the solution, let's quickly recap what exponential inequalities are all about. In essence, an exponential inequality is an inequality where the variable appears in the exponent. Solving these inequalities involves manipulating the expressions to isolate the variable while paying close attention to the base of the exponential terms. Key to this is understanding how exponential functions behave – particularly whether they are increasing or decreasing, which depends on whether the base is greater than 1 or between 0 and 1. When the base is greater than 1 (like our case where we will be working with powers of 3), the function is increasing, meaning that if a^x < a^y, then x < y. This property is super important, so keep it in mind!

When solving exponential inequalities, your main goal is to get the same base on both sides of the inequality. Once you have the same base, you can then compare the exponents directly. Remember, if the base is between 0 and 1, the direction of the inequality sign flips when you compare the exponents. But since our base will be greater than 1, we don’t have to worry about that this time. This approach simplifies the problem significantly and allows us to solve for the unknown variable in a straightforward manner. So, the roadmap is clear: find a common base, rewrite the inequality in terms of that base, and then solve for x.

Also, remember your exponent rules! These rules are essential for simplifying and manipulating exponential expressions. For example, (a^m)n = a^(m*n) and 1/a^n = a^(-n). Knowing these rules inside and out will make your life so much easier when dealing with exponential inequalities. It's also a good idea to double-check your work, especially when dealing with negative exponents or fractions. A small mistake can easily throw off your entire solution. Accuracy and attention to detail are key!

Step 1: Expressing Both Sides with the Same Base

The first crucial step in solving the inequality 9^(3x+2) < 1/81^(2x+5) is to express both sides with the same base. Notice that both 9 and 81 are powers of 3. Specifically, 9 = 3^2 and 81 = 3^4. This is our ticket to simplifying the inequality.

Let's rewrite the inequality using base 3:

• 9^(3x+2) = (3²⁾(3x+2) = 3^(2(3x+2)) = 3^(6x+4)*
• 1/81^(2x+5) = 1/(3⁴⁾(2x+5) = 1/3^(4(2x+5)) = 1/3^(8x+20) = 3^-(8x+20) = 3^(-8x-20)*

So, our inequality now looks like this: 3^(6x+4) < 3^(-8x-20). See how much simpler that looks? By expressing both sides with the same base, we've transformed the problem into something much more manageable. This step highlights the power of recognizing common bases and using exponent rules to simplify complex expressions.

Identifying common bases is not always obvious, but with practice, you'll get better at spotting them. Look for numbers that are powers of smaller primes like 2, 3, 5, and 7. Being comfortable with powers of these numbers will significantly speed up your problem-solving process. Remember, the key to success in math is often recognizing patterns and applying the right tools, and in this case, the right tool is expressing everything in terms of a common base.

Step 2: Comparing the Exponents

Now that we have the same base on both sides of the inequality – 3^(6x+4) < 3^(-8x-20) – we can directly compare the exponents. Since the base (3) is greater than 1, the exponential function is increasing. This means that if 3^(6x+4) < 3^(-8x-20), then 6x + 4 < -8x - 20. This step is where the magic happens. By comparing the exponents, we've transformed an exponential inequality into a simple linear inequality.

It's super important to remember that this direct comparison of exponents only works when you have the same base on both sides. And remember, if the base were between 0 and 1, you'd have to flip the inequality sign when comparing the exponents. But in our case, since the base is 3, we don't need to worry about that.

Think of this step as "peeling away" the exponential part of the problem to reveal the underlying linear relationship between the exponents. This technique is a cornerstone of solving exponential inequalities, and it's what makes these problems solvable. So, once you've got the same base, comparing the exponents is the next logical step. Don't forget to double-check that your base is greater than 1 to ensure you don't need to flip the inequality sign!

Step 3: Solving the Linear Inequality

We've successfully transformed our exponential inequality into a linear inequality: 6x + 4 < -8x - 20. Now, it's time to solve for x. This involves isolating x on one side of the inequality. Let's add 8x to both sides: 6x + 8x + 4 < -8x + 8x - 20 which simplifies to 14x + 4 < -20. Next, subtract 4 from both sides: 14x + 4 - 4 < -20 - 4 which simplifies to 14x < -24. Finally, divide both sides by 14: 14x / 14 < -24 / 14 which simplifies to x < -24/14. We can further simplify the fraction by dividing both the numerator and denominator by 2, resulting in x < -12/7.

And there you have it! The solution to the linear inequality is x < -12/7. This means that any value of x less than -12/7 will satisfy the original exponential inequality. Solving linear inequalities is a fundamental skill in algebra, and it's essential for tackling more complex problems like this one. Remember to perform the same operations on both sides of the inequality to maintain balance and isolate the variable you're solving for. And always double-check your work to ensure you haven't made any arithmetic errors.

In summary, solving the linear inequality involves a series of algebraic manipulations: adding, subtracting, multiplying, and dividing. The goal is always to isolate the variable on one side of the inequality to determine the range of values that satisfy the condition. With practice, you'll become more comfortable and confident in your ability to solve these types of inequalities.

Step 4: Stating the Solution

After all that work, we've arrived at the solution! The solution to the inequality 9^(3x+2) < 1/81^(2x+5) is x < -12/7. This means that any value of x that is less than -12/7 will satisfy the original inequality. To put it another way, if you plug in any number smaller than -12/7 into the original inequality, the left side will always be less than the right side.

It's often helpful to visualize the solution on a number line. Draw a number line and mark -12/7 on it. Since the inequality is "less than" and not "less than or equal to," we use an open circle at -12/7 to indicate that -12/7 itself is not included in the solution. Then, shade the portion of the number line to the left of -12/7 to represent all the values of x that satisfy the inequality. This visual representation can make it easier to understand the solution set.

Also, it's always a good idea to check your solution by plugging in a value from the solution set into the original inequality. For example, let's try x = -2 (which is less than -12/7). Plugging this into the original inequality, we get: 9^(3(-2)+2) < 1/81^(2(-2)+5)** which simplifies to 9^(-4) < 1/81^(1), which is 1/6561 < 1/81, which is true. This confirms that our solution is correct. Similarly, you can test a value outside the solution set (e.g., x = 0) to verify that it does not satisfy the original inequality. This process helps to ensure that you haven't made any mistakes along the way.

Conclusion

So there you have it, guys! We've successfully solved the exponential inequality 9^(3x+2) < 1/81^(2x+5). Remember the key steps:

1. Express both sides with the same base.
2. Compare the exponents.
3. Solve the resulting linear inequality.
4. State the solution.

With practice, you'll become more comfortable with these types of problems. Keep practicing, and you'll be an exponential inequality master in no time! Good luck, and happy solving!