Calculate The Sum Of Elements In Set A Where A Equals X+2 Divided By X Belongs To Natural Numbers And 11 Is Less Than Or Equal To 3x Plus 2 Which Is Less Than Or Equal To 20

Hey there, math enthusiasts! Today, we're diving into a fun little problem involving sets, inequalities, and a bit of arithmetic. We've got a set A defined by a specific condition, and our mission, should we choose to accept it, is to calculate the sum of all the elements in this set. Sounds intriguing? Let's jump right in!

Understanding the Set A

Before we can even think about summing the elements, we need to figure out what those elements are. Our set A is defined as {x+2/ x ∈ ℕ ∧ 11 ≤ 3x + 2 ≤ 20}. Let's break this down piece by piece:

• x ∈ ℕ: This tells us that 'x' belongs to the set of natural numbers. Natural numbers are the positive whole numbers, starting from 1 (i.e., 1, 2, 3, and so on). No fractions, no decimals, just good old whole numbers.
• 11 ≤ 3x + 2 ≤ 20: This is an inequality that gives us a range for the values of 'x'. It essentially puts a constraint on which natural numbers can be used to form the elements of our set. This inequality is the key to unlocking the contents of set A. We need to find the values of x that satisfy this condition to determine the elements that belong to our set. Think of it as a filter, only allowing certain natural numbers to pass through and contribute to the elements of A.
• x+2: This is the formula for the elements of the set. Once we find the valid values of 'x', we'll add 2 to each of them to get the actual elements that make up set A. It's like a transformation step, taking the filtered x values and turning them into the members of our set.

In essence, we're looking for natural numbers 'x' that, when plugged into the expression 3x + 2, fall between 11 and 20 (inclusive). Once we've found those 'x' values, we add 2 to each to get the elements of set A. Now, let's roll up our sleeves and solve this inequality!

Solving the Inequality

To find the values of 'x' that satisfy the inequality 11 ≤ 3x + 2 ≤ 20, we need to isolate 'x' in the middle. We can do this by performing the same operations on all three parts of the inequality. Let's go step by step:

1. Subtract 2 from all parts: This gets rid of the '+ 2' term on the side with 'x'.

   11 - 2 ≤ 3x + 2 - 2 ≤ 20 - 2

   9 ≤ 3x ≤ 18

2. Divide all parts by 3: This isolates 'x' in the middle.

   9 / 3 ≤ 3x / 3 ≤ 18 / 3

   3 ≤ x ≤ 6

So, we've found that 'x' must be greater than or equal to 3 and less than or equal to 6. But remember, 'x' has to be a natural number! This means the possible values for 'x' are 3, 4, 5, and 6. These are the natural numbers that satisfy the inequality and will help us build our set A. These are the x values that passed through our filter, ready to be transformed into the elements of set A.

Constructing Set A

Now that we know the possible values of 'x', we can finally construct the elements of set A. Remember, the elements are given by the formula x + 2. Let's plug in our values of 'x':

• For x = 3, the element is 3 + 2 = 5
• For x = 4, the element is 4 + 2 = 6
• For x = 5, the element is 5 + 2 = 7
• For x = 6, the element is 6 + 2 = 8

Therefore, set A is {5, 6, 7, 8}. We've successfully identified all the members of our set! Now we're just one step away from our final answer. We've built our set A element by element, transforming each valid x value into its corresponding member. It's like crafting a recipe, where the x values are the ingredients, and the formula x + 2 is the cooking process that creates the final dish – our set A.

Calculating the Sum

Our final task is to calculate the sum of the elements in set A. This is the easy part! We simply add up all the elements we found:

Sum = 5 + 6 + 7 + 8 = 26

And there you have it! The sum of the elements in set A is 26. We've successfully navigated through the inequality, identified the elements, and calculated their sum. It's like reaching the summit of a mathematical mountain – a satisfying feeling of accomplishment!

Conclusion

We've successfully tackled this problem by breaking it down into smaller, manageable steps. We started by understanding the definition of set A, then we solved the inequality to find the valid values of 'x', constructed the set, and finally, calculated the sum of its elements. This problem demonstrates the power of combining different mathematical concepts – sets, inequalities, and arithmetic – to solve a single problem. So next time you encounter a similar challenge, remember the steps we took today, and you'll be well on your way to finding the solution! Keep exploring, keep learning, and keep having fun with math, guys!

In summary, we've learned how to decipher set notation, solve inequalities, and apply these concepts to calculate the sum of elements within a set. These are valuable skills that can be applied to a wide range of mathematical problems. The key takeaway is to approach complex problems systematically, breaking them down into smaller, more manageable steps. This not only makes the problem less daunting but also allows for a clearer understanding of the underlying concepts. Remember, math is not just about finding the right answer, it's about the journey of discovery and the satisfaction of solving a puzzle!

Keywords: set, natural numbers, inequality, sum, elements