Sum Of Natural Values In Domain Of √9-3x

Hey guys! Today, we're diving into a fun math problem where we need to figure out the sum of natural numbers that fit within the domain of a square root expression. Specifically, we're tackling the expression √(9 - 3x). This might sound a bit intimidating at first, but trust me, we'll break it down step-by-step and make it super easy to understand. So, grab your thinking caps, and let's get started!

Understanding the Domain of the Expression

Okay, first things first, what exactly is the domain of an expression? In simple terms, the domain refers to all the possible values of the variable (in our case, x) that will make the expression mathematically valid. Now, when we're dealing with square roots, there's a golden rule we need to remember: we can't take the square root of a negative number (at least not in the realm of real numbers). This is super important because it sets the stage for finding our domain. So, for the expression √(9 - 3x) to be valid, the value inside the square root, (9 - 3x), must be greater than or equal to zero. This is the key to solving our problem.

Let's translate this into a mathematical inequality: 9 - 3x ≥ 0. To solve this inequality, we need to isolate x. We can start by subtracting 9 from both sides, which gives us -3x ≥ -9. Now, here's another crucial point: when we divide or multiply both sides of an inequality by a negative number, we need to flip the inequality sign. So, dividing both sides by -3, we get x ≤ 3. What does this mean? It means that the domain of our expression includes all real numbers x that are less than or equal to 3. Basically, any number 3 or smaller will work in our square root expression without causing any mathematical chaos. But remember, we're not just looking for any number; we're specifically interested in the natural numbers within this domain. So, let's zoom in on those.

Identifying Natural Values

Alright, now that we've nailed down the domain, let's talk about natural numbers. What are they? Natural numbers are the positive whole numbers we use for counting – 1, 2, 3, and so on. Zero is usually excluded from the set of natural numbers (though some definitions might include it, but for our purposes, we'll stick to the positive ones). So, we need to identify which of these natural numbers fall within our domain, which we've already determined is x ≤ 3. Thinking about it, the natural numbers that are less than or equal to 3 are simply 1, 2, and 3. These are the only positive whole numbers that fit the bill. We've narrowed it down, guys! We're almost at the finish line. Now comes the fun part – adding them up.

Calculating the Sum

Okay, guys, the final step! We've identified the natural numbers within the domain of our expression as 1, 2, and 3. Now, all that's left to do is find their sum. This is super straightforward: 1 + 2 + 3 = 6. And there you have it! The sum of the natural values of the variable x within the domain of the expression √(9 - 3x) is 6. We did it!

So, to recap, we started by understanding the importance of the domain for square root expressions, ensuring we don't end up with the square root of a negative number. We then translated this into an inequality, solved for x, and determined that our domain is x ≤ 3. Next, we focused on natural numbers, identifying 1, 2, and 3 as the ones that fit within our domain. Finally, we added them all up to arrive at our answer: 6. See? Math can be fun and totally manageable when we break it down into smaller, digestible steps.

Let's tackle another math problem together, guys! This time, we're going to find the sum of natural numbers that fall within the domain of the expression √9-3x (the entire expression is under the square root). This is a classic problem that combines our understanding of square roots, inequalities, and natural numbers. So, let's put on our thinking caps and dive right in!

Determining the Expression's Domain

The first thing we need to do is figure out the domain of the expression √9-3x. Remember, the domain is the set of all possible values of x for which the expression gives us a real number result. When we're dealing with square roots, there's a key rule we absolutely must keep in mind: the value under the square root (the radicand) cannot be negative. If it is, we'll end up with imaginary numbers, which aren't what we're looking for in this case. This is a fundamental principle, so let's make sure we've got it nailed down! For the expression √9-3x to be valid, the radicand, which is 9-3x, must be greater than or equal to zero. In mathematical terms, this translates to the inequality 9-3x ≥ 0. This inequality is the key to unlocking the domain of our expression. It tells us the condition that x must satisfy to give us a real number result. So, now our task is to solve this inequality for x. Let's see how we can do that.

To solve the inequality 9-3x ≥ 0, we'll use some algebraic manipulation. First, we want to isolate the term with x in it. We can do this by subtracting 9 from both sides of the inequality. This gives us -3x ≥ -9. Now, we have a term with x on one side and a constant on the other. The next step is to get x by itself. To do this, we need to divide both sides of the inequality by -3. But here's a crucial point: when we divide or multiply an inequality by a negative number, we have to flip the inequality sign. This is a very important rule, so let's make sure we remember it! Dividing both sides by -3 and flipping the inequality sign, we get x ≤ 3. This is our solution! It tells us that the domain of the expression √9-3x consists of all real numbers x that are less than or equal to 3. In other words, any value of x that is 3 or smaller will make the expression valid. But the problem asks us specifically for natural values of x. So, let's narrow our focus to those.

Identifying Natural Number Solutions

Now that we've determined the domain of the expression, let's focus on the natural numbers within that domain. What are natural numbers, exactly? Well, they are the positive whole numbers: 1, 2, 3, 4, and so on. They're the numbers we use for counting. Zero is typically not considered a natural number (although some definitions may include it, for our purposes, we'll exclude it). So, we need to find the natural numbers that satisfy the condition x ≤ 3. This means we're looking for positive whole numbers that are less than or equal to 3. If we think about it, the natural numbers that fit this description are simply 1, 2, and 3. These are the only positive whole numbers that are within our domain. We've successfully narrowed down the possible values of x to just these three numbers. Now, the final step is to find the sum of these numbers. Are you ready to add them up?

Summing the Values

Alright, guys, we've reached the final part of our problem! We've identified the natural number values of x within the domain of the expression √9-3x as 1, 2, and 3. Now, all that's left to do is find the sum of these numbers. This is a straightforward addition problem: 1 + 2 + 3. Let's add them up: 1 + 2 equals 3, and 3 + 3 equals 6. So, the sum of the natural values of x in the domain of our expression is 6! We've solved it! This means that if we plug in 1, 2, or 3 for x in the expression √9-3x, we'll get a real number result. And the sum of these natural number inputs is 6. Awesome job, guys! We've tackled this problem step by step, from determining the domain to identifying natural numbers and finally calculating their sum.

So, to recap, we started by understanding the crucial rule that the radicand (the value under the square root) must be non-negative. We translated this into the inequality 9-3x ≥ 0 and solved it to find the domain: x ≤ 3. Then, we focused on natural numbers, which are the positive whole numbers, and identified 1, 2, and 3 as the ones that fall within our domain. Finally, we added these numbers together to get our answer: 6. See how breaking down a problem into smaller, manageable steps can make it much easier to solve? Keep this in mind as you tackle other math challenges! You got this!

Hey everyone! Let's dive into another interesting math problem. We're going to find the sum of all the natural numbers that fit within the domain of the expression √9-3x (with the entire expression under the radical). This problem combines concepts from algebra, including inequalities and domain restrictions, with our knowledge of number systems, specifically natural numbers. So, grab your pencils and paper, and let's work through it together!

Establishing the Expression's Domain

Our first task is to determine the domain of the expression √9-3x. As we know, the domain of a function or expression is the set of all possible input values (in this case, x values) that will produce a valid output. When we're dealing with square roots, there's a fundamental restriction we need to be aware of: we cannot take the square root of a negative number (within the realm of real numbers, at least). This is a critical concept for this problem, so let's make sure we're all on the same page. For the expression √9-3x to be defined in the real number system, the value under the square root, which is 9-3x, must be greater than or equal to zero. This is because the square root of a non-negative number is a real number, while the square root of a negative number is an imaginary number. So, we need to ensure that 9-3x is not negative. We can express this requirement as an inequality: 9-3x ≥ 0. This inequality is the key to finding the domain of our expression. It tells us the condition that x must satisfy to give us a real number output. Now, our goal is to solve this inequality for x. How can we do that? Let's explore the steps.

To solve the inequality 9-3x ≥ 0, we'll use some algebraic techniques. Our aim is to isolate x on one side of the inequality. We can start by subtracting 9 from both sides. This gives us -3x ≥ -9. Now, we have the term with x on one side and a constant on the other. To get x by itself, we need to divide both sides by -3. But here's a very important point to remember: when we divide or multiply an inequality by a negative number, we must reverse the direction of the inequality sign. This is a crucial rule to avoid making mistakes, so let's make sure we've internalized it. Dividing both sides by -3 and reversing the inequality sign, we get x ≤ 3. This is the solution to our inequality! It tells us that the domain of the expression √9-3x is the set of all real numbers x that are less than or equal to 3. In other words, any value of x that is 3 or smaller will make the expression valid. However, the problem asks us to find the sum of the natural values of x within this domain. So, let's shift our focus to natural numbers.

Identifying Natural Number Solutions

Now that we've determined the domain of the expression, let's pinpoint the natural numbers that fall within it. What exactly are natural numbers? They are the positive whole numbers: 1, 2, 3, 4, and so on. They're the numbers we use for counting. Zero is generally not included in the set of natural numbers (though some definitions may include it, for our purposes, we'll exclude it). So, we're looking for natural numbers that satisfy the condition x ≤ 3. This means we need to find the positive whole numbers that are less than or equal to 3. If we think about it, the natural numbers that meet this criterion are simply 1, 2, and 3. These are the only positive whole numbers that are within our domain. We've successfully narrowed down the possible values of x to just these three numbers. Now, the final step is to calculate the sum of these natural numbers. Ready to add them up?

Summing the Values

Okay, everyone, we're at the final stage of our problem! We've identified the natural number values of x that belong to the domain of the expression √9-3x as 1, 2, and 3. Now, all that's left to do is find the sum of these numbers. This is a simple addition problem: 1 + 2 + 3. Let's add them together: 1 + 2 equals 3, and 3 + 3 equals 6. Therefore, the sum of the natural values of x within the domain of our expression is 6! We've successfully solved the problem! This means that if we substitute 1, 2, or 3 for x in the expression √9-3x, we will obtain a real number result. And the sum of these natural number inputs is 6. Excellent work, everyone! We've tackled this problem step-by-step, from determining the domain to identifying natural numbers and finally calculating their sum.

To summarize, we began by understanding the crucial rule that the radicand (the expression under the square root) must be non-negative. We translated this into the inequality 9-3x ≥ 0 and solved it to find the domain: x ≤ 3. Next, we focused on natural numbers, which are the positive whole numbers, and identified 1, 2, and 3 as the ones that fall within our domain. Lastly, we added these numbers together to get our answer: 6. Notice how breaking down a complex problem into smaller, more manageable steps can make it much easier to solve? Keep this approach in mind as you encounter other math challenges! You've got this!