Hey math enthusiasts! Today, we're diving deep into the world of inequalities, specifically tackling the problem: 9^(3-x) ≥ 1/(81^(2x+5)). Don't worry, it might look a little intimidating at first, but trust me, we'll break it down into easy-to-understand steps. By the end of this guide, you'll be able to confidently solve this kind of problem. This is a journey, and we'll take it together, step-by-step. Get ready to flex those math muscles and unlock a deeper understanding of inequalities.

Let's start by understanding what an inequality even is. In simple terms, an inequality is a mathematical statement that compares two expressions using symbols like greater than ( > ), less than ( < ), greater than or equal to ( ≥ ), or less than or equal to ( ≤ ). Unlike equations, which state that two expressions are equal, inequalities show the relationship where one expression is not necessarily equal to the other. Now, the problem at hand, 9^(3-x) ≥ 1/(81^(2x+5)), asks us to find all the values of 'x' that make the left-hand side greater than or equal to the right-hand side. This is where the fun begins. We'll be using the properties of exponents and inequalities to systematically solve this problem. The core idea is to manipulate the expressions to get the same base, which will allow us to compare the exponents directly. So, buckle up, and let's unravel this mathematical puzzle together. We'll ensure that you not only solve the problem correctly but also understand the principles behind it.

Step 1: Expressing Both Sides with the Same Base

Alright, guys, the first crucial step in solving this inequality is to get both sides of the equation to have the same base. You'll notice that we have a '9' on one side and an '81' on the other. But hey, both 9 and 81 can be expressed as powers of 3! Remember, this is the key to simplifying the problem. We can rewrite 9 as 3^2, and 81 as 3^4. So, let's substitute these values into our original inequality, shall we? Replace 9 with 3^2 and 81 with 3^4. The inequality now becomes (3²⁾(3-x) ≥ 1/((3⁴⁾(2x+5)).

Now, how to make the base the same, you ask? We use the properties of exponents to simplify the expressions. When you raise a power to another power, you multiply the exponents. Let's apply that to both sides. On the left side, (3²⁾(3-x) becomes 3^(2*(3-x)) or 3^(6-2x). On the right side, ((3⁴⁾(2x+5)) becomes 3^(4*(2x+5)) or 3^(8x+20). Also, remember that anything in the denominator can be expressed as a negative exponent in the numerator, so 1/(3^(8x+20)) becomes 3^-(8x+20) or 3^(-8x-20). The inequality is now 3^(6-2x) ≥ 3^(-8x-20). Isn't it cool how things are simplifying? We're making progress. Now that we have the same base on both sides, the next step is super important, it's about comparing the exponents. We will move to the next step, where we compare exponents of both sides and solve them.

Why is the same base important?

Because of the fundamental properties of exponential functions. When the bases are the same, and the inequality symbol is preserved, you can directly compare the exponents. It's like comparing apples to apples, instead of apples to oranges. This property is what allows us to simplify the problem to a point where we're dealing with a linear inequality, which is way easier to solve.

Step 2: Comparing the Exponents and Solving the Inequality

Now that we've got the same base (3) on both sides of the inequality, we can directly compare the exponents. Here’s the rule: if the base is greater than 1, like our base 3, then the inequality sign remains the same when we compare the exponents. So, we can rewrite our inequality 3^(6-2x) ≥ 3^(-8x-20) as 6 - 2x ≥ -8x - 20.

Let’s bring this to simpler form! Next, we need to solve this linear inequality for 'x'. First, let's get all the 'x' terms on one side and the constants on the other. Add 8x to both sides to get rid of the -8x on the right, resulting in 6 - 2x + 8x ≥ -20. Simplify to get 6 + 6x ≥ -20. Now, subtract 6 from both sides to isolate the term with 'x', which gives us 6x ≥ -26. Finally, divide both sides by 6 to solve for 'x'. This gives us x ≥ -26/6, which simplifies to x ≥ -13/3. Congratulations, you've solved for 'x'! It means that any value of 'x' greater than or equal to -13/3 satisfies the original inequality.

Understanding the Solution

The solution, x ≥ -13/3, means that any number equal to or greater than -13/3 will make the original inequality true. This is the range of values for which the inequality holds. You can test a few values to see it in action. For example, if you plug in x = 0 (which is greater than -13/3), you’ll find that the left side of the original inequality is greater than or equal to the right side. Similarly, plugging in x = -4 (also greater than -13/3) will also satisfy the inequality.

Step 3: Expressing the Solution in Interval Notation

Okay, math wizards, we have found the solution to our inequality: x ≥ -13/3. Now, let’s express this in a more formal way: interval notation. Interval notation is a concise way to represent all the values that satisfy an inequality. In this case, since x can be any number greater than or equal to -13/3, we use a square bracket to indicate that -13/3 is included in the solution, and we go all the way to positive infinity. Remember, infinity is never included, so it always gets a parenthesis. Our solution in interval notation is [-13/3, ∞). This notation tells us that the solution includes all real numbers starting from -13/3 (inclusive) and extending to infinity. It's a neat way to summarize our findings!

Let's understand this a little further. When the inequality includes the “equal to” part (≥ or ≤), as in our case (≥), we use a square bracket “[“ to show that the endpoint is included. If the inequality does not include the “equal to” part (> or <), we would use a parenthesis “(“ to show that the endpoint is not included. The infinity symbol (∞) always gets a parenthesis because infinity is not a specific number.

Visualization

You could also visualize this solution on a number line. Draw a number line, mark -13/3 on it, and draw a closed circle (because -13/3 is included). Then, shade the line to the right of -13/3, indicating all values greater than -13/3. This number line provides a visual representation of all the values of x that satisfy the inequality.

Step 4: Verification and Conclusion

Alright, folks, before we pat ourselves on the back, it's always a good idea to verify our solution. Let's do a quick check to make sure we're on the right track. Pick a number that's greater than -13/3, like x = 0. Substitute it into the original inequality: 9^(3-0) ≥ 1/(81^(2*0+5)). Simplify to get 9^3 ≥ 1/81^5. Calculate both sides: 729 ≥ 1/3486784401. This is true! The left side is indeed greater than the right side. This confirms our solution.

Now, let's take a moment to recap what we did and why. We started with the inequality 9^(3-x) ≥ 1/(81^(2x+5)). We then expressed both sides with the same base, which was 3. After that, we compared the exponents and solved the resulting linear inequality. Finally, we expressed our solution in interval notation: [-13/3, ∞). This entire process highlights the importance of understanding exponents, inequalities, and how to manipulate them to solve complex problems. Congratulations on reaching the end! You've successfully conquered this inequality.

Final Thoughts

Solving inequalities may seem challenging at first, but with practice, it becomes easier. Remember to always double-check your work and to express your solution clearly, either in interval notation or on a number line. Mathematics isn't just about getting the right answer; it's about understanding the process and the underlying principles. Keep practicing, keep learning, and keep asking questions. Until next time, keep those math skills sharp! We hope this step-by-step guide has empowered you to tackle similar problems with confidence. Keep exploring the world of mathematics and stay curious. You've got this!