Solving Inequalities Finding The Value Of X In 2x - 3 > 11 - 5x

Hey guys! Today, we're diving into a fun little math problem that involves inequalities. Inequalities might seem a bit tricky at first, but don't worry, we'll break it down step by step so it's super easy to understand. Our main goal here is to find the value of x that fits into the solution set of the inequality 2x - 3 > 11 - 5x. Sounds like a mission? Absolutely! So, let's jump right in and get those brain muscles flexing!

Understanding Inequalities

Before we tackle the specific problem, let's quickly chat about what inequalities are all about. Think of inequalities as mathematical statements that compare two values, but instead of saying they are equal (=), they tell us if one value is greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) another value. In our case, we have the 'greater than' sign (>), which means we're looking for all the values of x that make the left side of the inequality bigger than the right side. Inequalities are super useful in real life too. Imagine you're saving up for a new gadget, and you need to make sure your savings are greater than the price tag. That's an inequality in action!

Now, let's look at the anatomy of our inequality: 2x - 3 > 11 - 5x. We've got x hanging out on both sides, some constants (those are the numbers without any x attached), and that all-important 'greater than' sign. Our mission, should we choose to accept it (and we do!), is to isolate x on one side of the inequality so we can figure out what values make the statement true. We'll use some algebraic techniques, which are just fancy words for mathematical moves, to get x all by itself. Think of it like solving a puzzle, where each step gets us closer to the final answer. And remember, inequalities are not as scary as they might seem. With a little bit of practice, you'll be solving them like a pro in no time! So, let's roll up our sleeves and get started on the first step of our adventure.

Solving the Inequality: Step-by-Step

Alright, let's get our hands dirty and solve this inequality! Our inequality is 2x - 3 > 11 - 5x. The first thing we want to do is gather all the x terms on one side of the inequality. It's like herding cats, but with mathematical terms! A neat trick is to add 5x to both sides. Why? Because it cancels out the -5x on the right side, leaving us with just a number there. This is a crucial step because it simplifies the inequality and brings us closer to isolating x. Remember, what we do to one side, we gotta do to the other. It's the golden rule of algebra! So, adding 5x to both sides gives us:

2x - 3 + 5x > 11 - 5x + 5x

This simplifies to:

7x - 3 > 11

See? We're already making progress! Now, we want to get rid of that pesky -3 on the left side. To do that, we'll add 3 to both sides. Adding the same number to both sides keeps the inequality balanced, just like adding equal weights to both sides of a seesaw. So, let's add 3 to both sides:

7x - 3 + 3 > 11 + 3

This simplifies to:

7x > 14

We're almost there, guys! Now, the final step to isolate x is to divide both sides by 7. Division is the opposite of multiplication, and since x is being multiplied by 7, we need to divide to undo that. So, let's divide both sides by 7:

7x / 7 > 14 / 7

This simplifies to:

x > 2

Woohoo! We've solved it! This means that x is greater than 2. Any number bigger than 2 will make the original inequality true. Now, let's check out our answer choices and see which one fits the bill.

Checking the Answer Choices

Now that we've cracked the code and found that x > 2, it's time to play detective and see which of the given answer choices fits our solution. We have four suspects: A. -3, B. 0, C. 2, and D. 4. Our mission is to find the one that's greater than 2. It's like a mathematical version of 'Where's Waldo?', but instead of a stripey shirt, we're looking for a number that fits our inequality.

Let's start with A. -3. Is -3 greater than 2? Nope, it's way down in the negative territory, so we can rule out A. Next up is B. 0. Is 0 greater than 2? Nah, 0 is less than 2, so B is also off the list. Now we come to C. 2. Is 2 greater than 2? Hmm, this one's a bit tricky. 2 is equal to 2, but our inequality says x has to be greater than 2, not equal to. So, C is not our culprit. Finally, we have D. 4. Is 4 greater than 2? Bingo! 4 is indeed bigger than 2, so it fits our solution perfectly. It's like finding the missing puzzle piece! We've caught our answer choice, and it's D. 4.

So, to recap, we solved the inequality 2x - 3 > 11 - 5x, and we found that x > 2. Then, we checked the answer choices and discovered that 4 is the only one that's greater than 2. This whole process is like a mathematical treasure hunt, where we follow the clues (the steps of solving the inequality) to find the hidden treasure (the correct answer). And just like any treasure hunt, it feels awesome when you finally find the prize! So, high-fives all around for cracking this one!

Conclusion: The Value of x

Alright guys, we've reached the end of our mathematical journey for today, and what a journey it has been! We started with a seemingly complex inequality, 2x - 3 > 11 - 5x, and through the power of step-by-step problem-solving, we've not only found the answer but also understood the underlying concepts. Remember, solving inequalities is all about isolating x, and we did that by adding terms to both sides, simplifying, and finally dividing. It's like building a tower, where each step is crucial for the tower to stand tall.

We discovered that x must be greater than 2 to satisfy the inequality. This means any number bigger than 2 will work. And when we checked the answer choices, we pinpointed D. 4 as the correct solution. This process highlights the importance of not just solving the inequality but also understanding what the solution means. It's not just about getting a number; it's about understanding the range of values that make the inequality true.

So, the final answer to our question is D. 4. But more importantly, we've gained valuable skills in solving inequalities, which is a fundamental concept in algebra and mathematics. Inequalities pop up in various real-world scenarios, from budgeting and finance to science and engineering. By mastering the art of solving inequalities, we're equipping ourselves with a powerful tool that can help us make informed decisions and tackle problems in various aspects of life. Keep practicing, keep exploring, and remember that every math problem is just a puzzle waiting to be solved! And who knows, maybe the next inequality you solve will help you unlock a real-world treasure!