Selesaikan Pertidaksamaan: 3(2x-3) ≥ 5(x+1)
Hey guys! Today we're diving into a super common math problem that often pops up in high school and even some college courses: solving inequalities. Specifically, we're going to tackle the inequality 3(2x-3) ≥ 5(x+1) and find its solution set within a given set of numbers, which are {11, 12, 13, 14, 15}. It might sound a bit intimidating at first, but trust me, it's all about breaking it down step-by-step. We'll go through the process of simplifying the inequality, isolating the variable 'x', and then checking which of the provided numbers actually satisfy the condition. This is a fundamental skill in algebra, and once you get the hang of it, you'll be able to solve all sorts of similar problems with confidence. So, grab your notebooks, maybe a coffee or your favorite drink, and let's get this math party started! We're not just aiming to find the answer; we're aiming to understand why it's the answer. This deeper understanding will make you a math whiz in no time, guys. Let's make math fun and accessible for everyone!
Understanding the Inequality
Alright, let's kick things off by really understanding what we're dealing with. The inequality 3(2x-3) ≥ 5(x+1) is our main player. The symbol '≥' means 'greater than or equal to'. So, we're looking for values of 'x' that make the left side of the equation either larger than the right side, or exactly equal to it. Think of it like a scale; we want the left side to be heavier or balanced with the right side. The variable 'x' is what we need to solve for. It's like a mystery number, and our job is to find out what it could be. But here's the twist: we're not just looking for any 'x'. We have a specific set of possible values for 'x' to test: the set {11, 12, 13, 14, 15}. This is called the domain or universal set for our problem. It means we only care about whether these specific numbers make the inequality true. This makes our job a bit easier because instead of an infinite number of possibilities, we have a finite list to check. So, our goal is to figure out which numbers from that list – 11, 12, 13, 14, or 15 – make the statement "3 times (2 times x minus 3) is greater than or equal to 5 times (x plus 1)" true. It’s like a treasure hunt where the treasure is the 'x' that fits the bill! We'll need to use our algebra skills to simplify both sides of the inequality first. This involves using the distributive property, which is a fancy way of saying we multiply the number outside the parentheses by each term inside. Don't worry, we'll walk through it together, step by step.
Step-by-Step Solution of the Inequality
Now, let's get down to business and solve this inequality, 3(2x-3) ≥ 5(x+1). The first thing we want to do is simplify both sides by using the distributive property. On the left side, we multiply 3 by both 2x and -3. So, 3 * 2x gives us 6x, and 3 * -3 gives us -9. The left side becomes 6x - 9. On the right side, we multiply 5 by both x and 1. So, 5 * x is 5x, and 5 * 1 is 5. The right side becomes 5x + 5. Now our inequality looks like this: 6x - 9 ≥ 5x + 5. Our next goal is to get all the 'x' terms on one side and all the constant numbers on the other. Let's subtract 5x from both sides. This will move the 'x' term from the right to the left. So, (6x - 5x) - 9 ≥ (5x - 5x) + 5, which simplifies to x - 9 ≥ 5. Now, we need to get rid of that -9 on the left side. We can do this by adding 9 to both sides. So, x - 9 + 9 ≥ 5 + 9, which gives us x ≥ 14. Wow, look at that! We've simplified the original complex-looking inequality into a super simple one: x must be greater than or equal to 14. This is the general solution to the inequality. It means any number that is 14 or larger will satisfy the original inequality. Pretty neat, right? This process of simplification is key in algebra, and it's used for equations and inequalities alike. Remember to always perform the same operation on both sides to keep the inequality balanced. Mistakes often happen when we forget this rule or when we make errors in arithmetic, especially with negative numbers. So, double-checking your work is super important, guys!
Testing the Solution Set
So, we've found our simplified inequality: x ≥ 14. This means 'x' has to be 14 or any number bigger than 14 to make the original inequality 3(2x-3) ≥ 5(x+1) true. But remember, we're not just looking for any number; we have a specific set of numbers to check: {11, 12, 13, 14, 15}. This is our domain, and we need to see which of these numbers fit our condition x ≥ 14. Let's go through them one by one, like detectives!
- Is 11 ≥ 14? Nope, 11 is less than 14. So, 11 is not part of our solution set.
- Is 12 ≥ 14? Again, no. 12 is smaller than 14.
- Is 13 ≥ 14? Still no. 13 is also less than 14.
- Is 14 ≥ 14? Yes! 14 is equal to 14, and our condition is 'greater than or equal to'. So, 14 is part of our solution set.
- Is 15 ≥ 14? Absolutely! 15 is greater than 14. So, 15 is also part of our solution set.
Therefore, the numbers from the set {11, 12, 13, 14, 15} that satisfy the inequality x ≥ 14 are 14 and 15. This means the himpunan penyelesaian (solution set) for the inequality 3(2x-3) ≥ 5(x+1) within the given domain is {14, 15}. It's like we filtered the original list and only kept the numbers that passed the test. This process of testing values is crucial, especially when you're working with inequalities over a specific finite set. It ensures you're providing the exact answer required by the problem. Always remember to refer back to the original problem's constraints. We weren't asked for all possible real numbers greater than or equal to 14; we were asked for which numbers from that specific list work. This careful attention to detail is what separates a good math student from a great one, guys! Keep up the awesome work!
Why This Matters: Real-World Applications
Okay, so we just solved an inequality and found a solution set. You might be thinking, "Cool math problem, but why does this matter in the real world?" That's a totally fair question, guys! Inequalities, like the one we just tackled, 3(2x-3) ≥ 5(x+1), are actually super useful in tons of real-life situations. Think about budgeting. Let's say you have a maximum amount of money you can spend, say $50. If 'x' represents the cost of an item, you might have an inequality like x ≤ 50, meaning the cost must be less than or equal to $50. Or maybe you need to achieve a certain score to pass a test. If 'S' is your score and the passing score is 70, then your inequality is S ≥ 70. This is exactly what we've been working with! Businesses use inequalities all the time for things like profit margins, production limits, and resource allocation. For example, a company might want its profit (P) to be at least $10,000, so P ≥ 10,000. Or they might have a limited number of raw materials, say 'm' units, and each product requires 2 units, so 2 * (number of products) ≤ m. Even in science, inequalities are used to describe ranges of conditions, like temperature or pressure limits for an experiment. They help define acceptable ranges and boundaries. Our problem, while using specific numbers, demonstrates the core concept: finding values that satisfy a certain condition. Whether it's deciding if you have enough money for groceries (your spending ≤ your budget) or checking if a product meets quality standards (measurement ≥ minimum requirement), inequalities are the mathematical language we use to express these limits and requirements. So, the next time you're solving an inequality, remember you're learning a practical skill that helps make decisions and set boundaries in all sorts of scenarios. Pretty cool, huh?
Conclusion: Mastering Inequalities
We've successfully navigated the world of inequalities, tackling 3(2x-3) ≥ 5(x+1) and finding its solution set within 11, 12, 13, 14, 15}. We learned that by applying the distributive property and basic algebraic manipulations, we simplified the inequality to x ≥ 14. Then, we diligently tested each number from our given set against this simplified condition. The numbers that met the criteria, 14 and 15, formed our final solution set**. This journey wasn't just about finding an answer; it was about understanding the process. We saw how to simplify, isolate the variable, and crucially, how to test values within a specific domain. Remember, guys, practice is key! The more you work through problems like this, the more comfortable and confident you'll become. Don't be afraid to go back and re-read steps, double-check your calculations, or even try solving the inequality again from scratch. Math is a skill that builds over time, and every problem you solve is a step forward. Keep exploring, keep questioning, and most importantly, keep enjoying the process of learning. You've got this!