Solving X + 2 ≤ X/3 + 3 Find Prime Number Solutions
Introduction
In this article, we will delve into the process of solving the given inequality, x + 2 ≤ x/3 + 3, and subsequently determine the solution set for x under the constraint that x is a prime number. This involves algebraic manipulation of the inequality to isolate x, followed by identifying the prime numbers within the solution range. Understanding inequalities and prime numbers is crucial in various mathematical contexts, making this a valuable exercise in mathematical problem-solving.
Step-by-Step Solution
To find the solution set for x, we need to solve the inequality step-by-step. Our primary goal is to isolate x on one side of the inequality. Here’s how we can approach it:
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Start with the given inequality: x + 2 ≤ x/3 + 3. The first step in simplifying this inequality is to eliminate the fraction. We can do this by multiplying every term in the inequality by 3. This ensures that we maintain the balance of the inequality while removing the denominator.
- Multiplying throughout by 3, we get: 3(x + 2) ≤ 3(x/3 + 3).
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Distribute the multiplication: Next, we distribute the 3 on both sides of the inequality to remove the parentheses. This involves multiplying each term inside the parentheses by 3.
- This simplifies to: 3x + 6 ≤ x + 9. Now we have a linear inequality without fractions, which is easier to work with.
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Isolate x terms: The next step is to group the x terms on one side of the inequality and the constant terms on the other side. We can do this by subtracting x from both sides.
- Subtracting x from both sides, we get: 3x - x + 6 ≤ x - x + 9, which simplifies to 2x + 6 ≤ 9.
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Isolate the x term: Now, we need to isolate the term containing x. We can do this by subtracting 6 from both sides of the inequality.
- Subtracting 6 from both sides, we get: 2x + 6 - 6 ≤ 9 - 6, which simplifies to 2x ≤ 3.
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Solve for x: To find the values of x, we need to divide both sides of the inequality by 2. This will give us the range of possible values for x.
- Dividing both sides by 2, we get: 2x/2 ≤ 3/2, which simplifies to x ≤ 1.5.
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Consider the prime number constraint: The problem states that x must be a prime number. Prime numbers are numbers greater than 1 that have only two distinct positive divisors: 1 and themselves. The first few prime numbers are 2, 3, 5, 7, 11, and so on. However, since our solution must satisfy x ≤ 1.5, we need to identify the prime numbers that meet this condition.
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Identify prime numbers within the solution range: Prime numbers are whole numbers greater than 1 that have only two divisors: 1 and the number itself. The prime numbers less than or equal to 1.5 need to be identified.
Determining the Solution Set
Given that x ≤ 1.5 and x must be a prime number, we need to determine which prime numbers satisfy this condition. Prime numbers are integers greater than 1 that have no positive divisors other than 1 and themselves. Let's analyze the prime numbers in relation to our inequality:
- The first few prime numbers are 2, 3, 5, 7, 11, and so on.
- We are looking for prime numbers that are less than or equal to 1.5.
Looking at the list of prime numbers, we can see that there are no prime numbers less than or equal to 1.5. The smallest prime number is 2, which is greater than 1.5. Therefore, there are no prime numbers that satisfy the inequality x ≤ 1.5.
Final Answer
Therefore, based on our step-by-step solution, there are no prime numbers that satisfy the inequality x ≤ 1.5. This means the solution set for x is empty.
Solution Set for x
The solution set for x is the empty set, denoted by ∅. This indicates that there are no values of x that satisfy both the inequality x + 2 ≤ x/3 + 3 and the condition that x is a prime number.
Conclusion
In conclusion, by solving the inequality x + 2 ≤ x/3 + 3 and applying the constraint that x must be a prime number, we have determined that there are no solutions. The solution set is the empty set, ∅. This exercise highlights the importance of understanding both algebraic inequalities and number theory concepts, specifically prime numbers, in solving mathematical problems. The ability to manipulate inequalities and apply specific conditions, such as primality, is a valuable skill in mathematics.
This process of solving inequalities and applying constraints is fundamental in many areas of mathematics and its applications, including optimization problems, number theory, and cryptography. Understanding these concepts provides a strong foundation for tackling more complex mathematical challenges. By systematically working through the steps, we can arrive at accurate solutions and gain a deeper appreciation for the interplay between different mathematical principles.
