Solving Exponential Inequality 3^(2x+1) + 9 - 28 * 3^x > 0 Step-by-Step
Hey guys! Today, we're diving deep into the exciting world of exponential inequalities, specifically tackling the problem: 3^(2x+1) + 9 - 28 * 3^x > 0. Don't worry if it looks intimidating at first glance. We'll break it down step-by-step, making sure everyone understands the process. Think of this as a friendly chat about how to conquer these types of mathematical puzzles. We'll not only solve this particular inequality but also equip you with the skills to handle similar problems with confidence. So, grab your favorite beverage, get comfy, and let's embark on this mathematical adventure together!
Understanding Exponential Inequalities
Before we jump into the solution, let's quickly recap what exponential inequalities are all about. Exponential inequalities, at their core, are inequalities where the variable appears in the exponent. You know, like our friend 'x' hanging out up there in 3^(2x+1). These inequalities often involve exponential functions, which means the variable is part of the exponent. This contrasts with polynomial inequalities, where the variable is typically the base. For instance, x^2 + 2x - 3 > 0 is a polynomial inequality, while our problem, 3^(2x+1) + 9 - 28 * 3^x > 0, is an exponential inequality. What makes these inequalities interesting is that the exponential function can grow very rapidly, and this rapid growth adds a unique twist to how we solve them. Unlike linear or quadratic inequalities, where we can often isolate the variable directly, exponential inequalities usually require some clever algebraic manipulation and a bit of substitution to simplify them. The key here is recognizing the exponential form and understanding how the properties of exponents can be used to our advantage. We'll be using these properties extensively as we solve our example problem, so keep them in mind!
Rewriting the Inequality
Okay, let's get our hands dirty with the actual problem. Our initial exponential inequality is 3^(2x+1) + 9 - 28 * 3^x > 0. The first step, and a crucial one, is to rewrite the inequality in a more manageable form. This involves using the properties of exponents to break down the terms and make it easier to spot patterns. Remember that a^(m+n) is the same as a^m * a^n. Applying this to our first term, 3^(2x+1), we can rewrite it as 3^(2x) * 3^1, which simplifies to 3 * 3^(2x). Now, let's tackle 3^(2x). We can further rewrite this as (3x)2. This step is super important because it helps us see a structure that we can work with. Our inequality now looks like 3 * (3x)2 + 9 - 28 * 3^x > 0. Notice how we've managed to express everything in terms of 3^x? This is a big win! It sets us up perfectly for the next step: substitution. By rewriting the inequality in this form, we've transformed a seemingly complex exponential inequality into something that resembles a quadratic inequality, which we are much more familiar with solving. This is a common technique when dealing with exponential inequalities, and mastering it will make solving these problems a breeze. So, always be on the lookout for opportunities to rewrite the expression using exponent rules; it's your secret weapon in these situations.
Substitution: A Clever Trick
Now comes the fun part – substitution! This is where we introduce a new variable to simplify the inequality. Let's say y = 3^x. Why are we doing this? Well, by substituting, we can transform our exponential inequality into a quadratic inequality, which is something we know how to solve. Replacing every instance of 3^x with 'y' in our rewritten inequality, 3 * (3x)2 + 9 - 28 * 3^x > 0, we get 3y^2 - 28y + 9 > 0. See? Much simpler, right? This substitution is a classic technique in solving exponential equations and inequalities. It's like a magic trick that makes the problem much more approachable. Now we have a quadratic inequality that we can solve using standard methods. But remember, guys, the goal isn't just to solve for 'y'; we eventually need to find the values of 'x'. So, keep in mind that y = 3^x, and we'll come back to this relationship later. The beauty of substitution lies in its ability to transform complex problems into simpler ones. It's a powerful tool in your mathematical arsenal, and understanding when and how to use it is key to tackling a wide range of problems. So, embrace the power of substitution, and watch those tricky inequalities melt away!
Solving the Quadratic Inequality
With our substitution in place, we're now facing a quadratic inequality: 3y^2 - 28y + 9 > 0. Time to put our quadratic-solving hats on! The first step in solving any quadratic inequality is to find the roots of the corresponding quadratic equation. That means we need to solve 3y^2 - 28y + 9 = 0. There are several ways to do this: factoring, using the quadratic formula, or even completing the square. For this particular equation, factoring is a good option. We're looking for two numbers that multiply to 3 * 9 = 27 and add up to -28. Those numbers are -27 and -1. So, we can rewrite the middle term as -27y - y, giving us 3y^2 - 27y - y + 9 = 0. Now, we factor by grouping: 3y(y - 9) - 1(y - 9) = 0. This simplifies to (3y - 1)(y - 9) = 0. Setting each factor to zero, we find the roots y = 1/3 and y = 9. These roots are crucial because they divide the number line into intervals where the quadratic expression has a consistent sign (either positive or negative). To determine the solution to the inequality 3y^2 - 28y + 9 > 0, we need to test values from each interval in the original inequality. This will tell us which intervals satisfy the condition. Remember, solving quadratic inequalities is a fundamental skill in algebra, and it's essential for tackling more advanced problems, like our exponential inequality. So, mastering these techniques will definitely pay off in the long run!
Determining the Intervals and Testing Values
Now that we've found the roots of our quadratic equation (y = 1/3 and y = 9), it's time to determine the intervals and see where our inequality, 3y^2 - 28y + 9 > 0, holds true. The roots divide the number line into three intervals: (-∞, 1/3), (1/3, 9), and (9, ∞). To figure out whether the quadratic expression is positive or negative in each interval, we'll pick a test value from each and plug it into the inequality. Let's start with the interval (-∞, 1/3). A convenient test value here is y = 0. Substituting y = 0 into 3y^2 - 28y + 9, we get 3(0)^2 - 28(0) + 9 = 9, which is greater than 0. So, the inequality holds true in the interval (-∞, 1/3). Next, let's consider the interval (1/3, 9). A simple test value here is y = 1. Substituting y = 1, we get 3(1)^2 - 28(1) + 9 = 3 - 28 + 9 = -16, which is less than 0. Thus, the inequality does not hold in this interval. Finally, let's test the interval (9, ∞). We can use y = 10 as our test value. Substituting y = 10, we get 3(10)^2 - 28(10) + 9 = 300 - 280 + 9 = 29, which is greater than 0. So, the inequality holds true in the interval (9, ∞) as well. This interval testing method is a cornerstone of solving inequalities. It allows us to systematically determine the solution set by breaking the number line into manageable chunks and checking the sign of the expression within each chunk. By carefully selecting test values and evaluating the inequality, we can confidently identify the intervals that satisfy the given condition.
Back to the Original Variable: Solving for x
We've made great progress, guys! We've solved the quadratic inequality in terms of 'y', but remember, our original problem was in terms of 'x'. It's time to go back to the original variable and solve for 'x'. We know that y = 3^x, and we found that the solution to the quadratic inequality 3y^2 - 28y + 9 > 0 is y < 1/3 or y > 9. Now, we need to translate these inequalities back into terms of 'x'. So, we have two cases to consider: 3^x < 1/3 and 3^x > 9. Let's tackle the first one: 3^x < 1/3. We can rewrite 1/3 as 3^(-1). So, we have 3^x < 3^(-1). Since the base is the same (3), and it's greater than 1, we can directly compare the exponents: x < -1. Now, let's move on to the second case: 3^x > 9. We can rewrite 9 as 3^2. So, we have 3^x > 3^2. Again, since the base is the same and greater than 1, we can compare the exponents: x > 2. Therefore, the solution to our original exponential inequality is x < -1 or x > 2. This step of converting back to the original variable is crucial in any substitution-based problem. It ensures that we're answering the question that was actually asked. Always remember to reverse the substitution and express your final answer in terms of the original variables. We're almost there! Just a little more to go, and we'll have conquered this exponential inequality.
The Final Solution and Conclusion
Phew! We've reached the end of our journey. The final solution to the exponential inequality 3^(2x+1) + 9 - 28 * 3^x > 0 is x < -1 or x > 2. We arrived at this answer by carefully following a series of steps: rewriting the inequality, making a clever substitution, solving the resulting quadratic inequality, and finally, converting back to the original variable. This process highlights the power of algebraic manipulation and the importance of understanding the properties of exponents. Guys, remember, exponential inequalities might seem daunting at first, but with a systematic approach, they become much more manageable. The key is to break the problem down into smaller, more digestible steps. Look for opportunities to rewrite the expression, use substitution to simplify the problem, and don't forget to convert back to the original variable at the end. Solving exponential inequalities is not just about finding the right answer; it's about developing problem-solving skills that can be applied to a wide range of mathematical challenges. By mastering these techniques, you'll not only be able to tackle exponential inequalities with confidence but also enhance your overall mathematical prowess. So, keep practicing, keep exploring, and never be afraid to dive into a new mathematical adventure!
