Solving Inequalities When X Equals Square Root Of 151 Divided By 2
Hey everyone! Today, we're diving into a fun mathematical problem that involves inequalities and square roots. It might seem a bit daunting at first, but trust me, we'll break it down step by step. Our main goal is to figure out which inequality holds true when x equals the square root of 151 divided by 2. We've got some options to choose from, and we're going to explore the best way to solve this. So, grab your thinking caps, and let's get started!
Understanding the Problem: Which Inequality Holds True?
Okay, so the core of our problem revolves around finding the correct inequality for a given value of x. Specifically, we're given that $x = \frac{\sqrt{151}}{2}$, and we need to determine which of the following inequalities is true:
- A. $300 < x < 302$
- B. $75 < x < 76$
- C. $13 < x < 14$
- D. $6 < x < 7$
At first glance, this might look tricky. We've got a square root involved, and we need to compare it to whole numbers. But don't worry! The key here is to estimate the value of the square root and then divide by 2. We'll use a combination of estimation and logical reasoning to narrow down our options and find the correct answer. The journey to solve this isn't just about getting the right answer; it's also about understanding why that answer is correct. We'll explore the underlying mathematical concepts and strategies that make solving inequalities like this easier. So, let's dive deeper into how we can approach this problem.
Estimation is Key: Approximating √151
The first step in tackling this problem is to get a good estimate of the square root of 151. We know that 151 isn't a perfect square, meaning its square root will be a decimal number. So, how do we find a good approximation? Well, we can think about the perfect squares that are close to 151. Remember, perfect squares are numbers that result from squaring an integer (like 1, 4, 9, 16, etc.).
We know that $12^2 = 144$ and $13^2 = 169$. Since 151 falls between 144 and 169, we can deduce that $\sqrt{151}$ will be between 12 and 13. But closer to which? Since 151 is closer to 144 than it is to 169, we can reasonably estimate that $\sqrt{151}$ is a little bit more than 12. To get a more precise estimate, we can consider that 151 is 7 more than 144. This is less than halfway to 169 (which is 25 more than 144), suggesting that $\sqrt{151}$ is closer to 12 than to 12.5. For the purpose of this problem, an estimate of 12.3 or 12.4 should be sufficient. This skill of approximation is super useful, not just in math problems, but also in everyday situations where you need to make quick calculations. By understanding perfect squares and their relationships, we can confidently estimate square roots and make informed decisions. Remember, the goal isn't always to find the exact answer, but to find an answer that's close enough for our needs.
Dividing by 2: Finding the Value of x
Now that we have a good estimate for $\sqrt{151}$, which we approximated to be around 12.3 or 12.4, the next step is to divide this value by 2. Remember, our original problem states that $x = \frac{\sqrt{151}}{2}$. So, we need to calculate this value to figure out which inequality x satisfies. Let's take our estimated value of 12.3 and divide it by 2. This gives us 6.15. If we use our higher estimate of 12.4, dividing by 2 gives us 6.2. So, we can confidently say that x is somewhere between 6.15 and 6.2. This step is crucial because it bridges the gap between the square root and the actual value of x that we need to compare with the given inequalities. Dividing by 2 scales down the value, making it easier to see where x fits within the number line.
This simple division allows us to translate our approximation of the square root into an approximation of x, which is exactly what we need to solve the problem. Now, with this value of x in hand, we're much closer to identifying the correct inequality. We've transformed the problem from dealing with a square root to dealing with a more manageable decimal number. This kind of simplification is a common strategy in math problem-solving: breaking down complex problems into smaller, more digestible steps.
Matching x to the Inequalities: Which One Fits?
We've reached the final stage of our puzzle! We've estimated that $x = \frac{\sqrt{151}}{2}$ is approximately between 6.15 and 6.2. Now, we need to compare this value with the given inequalities to see which one holds true. Let's revisit the options:
- A. $300 < x < 302$
- B. $75 < x < 76$
- C. $13 < x < 14$
- D. $6 < x < 7$
By simply looking at these inequalities, we can eliminate options A, B, and C almost immediately. Our estimated value of x (around 6.15 to 6.2) is nowhere near the ranges specified in those inequalities. 300? 75? 13? No way! This is where the power of estimation truly shines. We didn't need to calculate the exact value of x to rule out these options.
Option D, $6 < x < 7$, looks much more promising. Our estimated value of x falls within this range. Since 6.15 and 6.2 are both greater than 6 and less than 7, this inequality holds true. Therefore, option D is the correct answer. This process of elimination, combined with our estimation skills, allowed us to efficiently solve the problem without getting bogged down in unnecessary calculations. This is a valuable technique for tackling multiple-choice questions, where time is often a factor.
Conclusion: The Correct Inequality
Alright, guys, we've cracked the code! After carefully estimating the square root of 151, dividing by 2, and comparing our result with the given inequalities, we've confidently determined that the correct answer is D. $6 < x < 7$. We've walked through the entire problem step-by-step, highlighting the importance of estimation and logical reasoning in solving mathematical challenges.
We started by understanding the problem, then we honed our estimation skills to approximate $\sqrt{151}$. We divided our estimate by 2 to find an approximate value for x, and finally, we compared this value to the given inequalities, eliminating options until we arrived at the correct answer. Remember, the journey is just as important as the destination. By breaking down the problem into smaller, manageable steps, we made the whole process less intimidating and more enjoyable. This approach can be applied to a wide range of mathematical problems, helping you build confidence and problem-solving skills. So, next time you encounter a tricky inequality, remember the techniques we've discussed here, and you'll be well on your way to finding the solution!
