Solve Math Inequality: A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into an intriguing problem that involves comparing fractions and algebraic expressions. It's like a puzzle where we need to decipher the relationship between a number and its transformations. We'll break down the problem step by step, making sure everyone, from math newbies to seasoned pros, can follow along. So, grab your thinking caps, and let's get started!

Decoding the Problem: Setting the Stage

Alright, let's get to the heart of the matter. Our mission is to solve this question: If one-third of a number, which we're calling x, is greater than five less than twice the number, which of the following inequalities must be true? This might sound like a mouthful, but don't worry, we're going to dissect it piece by piece. The key here is to translate the words into mathematical language. Think of it as learning a new dialect – the dialect of algebra!

The core of this problem lies in understanding how to represent mathematical relationships using symbols and equations. We're dealing with an unknown number, x, and several operations performed on it: taking one-third of it, doubling it, and subtracting five. The phrase "is greater than" is our signal to use an inequality symbol. Inequalities, unlike equations, show a range of possible values rather than a single solution. This adds a layer of complexity, but also a world of possibilities. Understanding inequalities is crucial in various real-world scenarios, from setting budgets to understanding scientific data. We will use our algebraic skills to decipher this inequality and determine the range of values that x can take.

The real trick is to break it down, right? So, let's start by identifying the key components of the problem. The phrase "one-third of a number, x" translates directly to (1/3)x or x/3. This is straightforward enough. But what about "five less than twice the number"? Here, the order of operations is important. We first double the number x, which gives us 2x, and then subtract five, resulting in 2x - 5. Now, we have two expressions: x/3 and 2x - 5. The problem states that x/3 is greater than 2*x - 5. This "greater than" relationship is what we'll express using the > symbol.

Before we jump into solving, let's pause for a moment and appreciate the power of this translation. We've taken a verbal statement, which can sometimes be ambiguous, and turned it into a precise mathematical expression. This is the essence of algebra – the art of representing relationships with symbols. It's like having a secret code that allows us to communicate complex ideas clearly and efficiently. By understanding this code, we can unlock a whole universe of mathematical problems. It is very important to master the basics before moving on to more complicated stuff. Once you get the basics, you will be able to solve harder problems.

Translating Words into Math: Crafting the Inequality

Alright, guys, let's get this show on the road and transform those words into a powerful mathematical inequality! Remember, we're deciphering the statement: "one-third of a number, x, is greater than five less than twice the number." We've already identified the key pieces: one-third of x is x/3, and five less than twice x is 2x - 5. Now, we just need to connect them using the "greater than" symbol, which is >.

So, putting it all together, we get the inequality: x/3 > 2x - 5. Boom! We've done it! We've successfully translated a verbal statement into a concise mathematical form. This is a huge step, because now we can use the tools of algebra to manipulate this inequality and solve for x. Think of it like building a bridge – we've laid the foundation, and now we can start constructing the rest of the structure. This skill of translating words into math is fundamental not just in algebra, but in all areas of mathematics and science. It's like learning the grammar of the universe!

Now that we have our inequality, x/3 > 2x - 5, let's take a closer look. It's important to understand what this inequality actually means. It's not just a bunch of symbols; it's a statement about the relationship between x/3 and 2x - 5. It says that for any value of x that satisfies this inequality, the value of x/3 will always be larger than the value of 2x - 5. This is a crucial concept to grasp, because it helps us visualize the solution. We're not looking for a single value of x; we're looking for a range of values that make this statement true. This range of values is what we call the solution set of the inequality.

Before we move on to solving, let's consider why inequalities are so important. In the real world, situations are rarely as clear-cut as equations. We often deal with constraints, limits, and ranges. For example, a budget might set a limit on spending, or a recipe might call for a certain range of ingredients. Inequalities allow us to model these situations mathematically. They provide a powerful tool for analyzing and solving problems that involve uncertainty and variability. It's like having a flexible ruler that can measure a range of possibilities, rather than a single, fixed point. So, by mastering inequalities, we're not just learning algebra; we're developing a valuable skill for tackling real-world challenges.

Solving the Inequality: Unraveling the Solution

Alright, time to roll up our sleeves and get down to the nitty-gritty of solving this inequality! We've got x/3 > 2x - 5, and our mission is to isolate x on one side of the inequality. It's like a detective story, where we're trying to uncover the hidden value of x. We'll use the rules of algebra to manipulate the inequality, making sure we maintain the integrity of the relationship. Remember, whatever we do to one side, we must do to the other.

Our first goal is to get rid of that fraction. Fractions can sometimes make things a bit messy, so let's multiply both sides of the inequality by 3. This will cancel out the denominator on the left side. So, we have 3 * (x/3) > 3 * (2x - 5). This simplifies to x > 6x - 15. See how much cleaner that looks? It's like decluttering your workspace before tackling a big project. By eliminating the fraction, we've made the inequality easier to work with.

Now, we want to get all the x terms on one side of the inequality. Let's subtract 6x from both sides: x - 6x > 6x - 15 - 6x. This gives us -5x > -15. We're getting closer! It's like following the breadcrumbs in a maze, each step brings us closer to the center. Notice that we're carefully applying the same operation to both sides, ensuring that the inequality remains balanced.

Here comes the crucial step. We need to divide both sides by -5 to isolate x. But hold on! There's a special rule we need to remember when dealing with inequalities: when we multiply or divide by a negative number, we must flip the inequality sign. This is a key concept that's easy to overlook, but it's essential for getting the correct solution. So, dividing both sides by -5 and flipping the sign, we get x < 3. Ta-da! We've solved for x!

Let's take a moment to appreciate what we've accomplished. We started with a complex verbal statement, translated it into an inequality, and then used the rules of algebra to solve for x. It's like cracking a code and revealing the hidden message. The solution, x < 3, tells us that any number less than 3 will satisfy the original condition. This is a range of values, not just a single number. It's like having a map that shows us all the possible destinations, rather than just one specific location. Understanding this concept of a solution set is crucial for working with inequalities. It's not just about finding the answer; it's about understanding the range of possibilities.

Deciphering the Solution: What Does x < 3 Really Mean?

Excellent work, team! We've successfully navigated the algebraic maze and arrived at our solution: x < 3. But what does this really mean? It's not just a bunch of symbols; it's a statement about the possible values of x that make our original inequality true. Think of it as a secret code that unlocks a range of possibilities. To truly understand this solution, we need to interpret it in the context of the original problem. It is important to understand the practical implications of the math you're doing.

x < 3 tells us that any number less than 3 will satisfy the condition that one-third of the number is greater than five less than twice the number. Let's try a few examples to see this in action. What if x is 2? One-third of 2 is 2/3, and five less than twice 2 is 2 * 2 - 5 = -1. Is 2/3 greater than -1? Yes, it is! So, 2 is a solution. How about x = 0? One-third of 0 is 0, and five less than twice 0 is -5. Is 0 greater than -5? Again, yes! So, 0 is also a solution. This is the beauty of inequalities – they give us a range of solutions, not just a single answer.

But what about numbers greater than or equal to 3? Let's try x = 3. One-third of 3 is 1, and five less than twice 3 is 2 * 3 - 5 = 1. Is 1 greater than 1? No, it's not. They're equal. So, 3 is not a solution. This confirms that our solution x < 3 is the correct boundary. Any number greater than 3 will also fail to satisfy the original inequality. It's like setting a limit – we know exactly where the valid values end and the invalid values begin.

Now, let's connect this back to the original question. The problem asked us to identify which of the given options is true. We've determined that x < 3 is the correct solution. This means that any of the options that are consistent with x being less than 3 are also true. For example, if one of the options was x < 4, that would also be true, because all numbers less than 3 are also less than 4. But the most precise answer is x < 3, as it gives us the exact upper limit of the solution set.

In the grand scheme of things, understanding how to interpret solutions like x < 3 is crucial for applying math to real-world problems. It's not enough to just solve the equation or inequality; we need to understand what the solution means in the context of the situation. This is where critical thinking and problem-solving skills come into play. It's like having a map and knowing how to read it, not just having the map itself. So, congratulations on mastering this skill! You're well on your way to becoming a math whiz!

Tying It All Together: Final Thoughts and Real-World Connections

We've reached the end of our algebraic adventure, guys! We started with a seemingly complex problem, dissected it, translated it into mathematical language, solved it, and then interpreted the solution. That's quite a journey! We've not only found the answer, but we've also deepened our understanding of inequalities and their power in representing real-world situations. It's like learning a new language and then using it to write a story. You've gained a new way to express and understand the world around you.

Let's recap the key steps we took. First, we carefully read the problem and identified the key phrases: "one-third of a number," "five less than twice the number," and "is greater than." We then translated these phrases into algebraic expressions and the appropriate inequality symbol. This step is crucial, because it sets the foundation for the rest of the solution. It's like building a strong framework for a house – everything else depends on it.

Next, we solved the inequality using the rules of algebra. We multiplied both sides by 3 to eliminate the fraction, simplified the expression, and then divided by -5, remembering to flip the inequality sign. This step required careful attention to detail and a solid understanding of algebraic principles. It's like navigating a complex maze – you need to know the rules and follow them carefully to reach the goal.

Finally, we interpreted the solution x < 3 in the context of the original problem. We tested values to see if they satisfied the inequality and discussed what the solution means in real-world terms. This step is often overlooked, but it's just as important as the mechanics of solving. It's like understanding the message behind the code – it's not enough to just break the code; you need to know what it means.

So, why is all of this important? Inequalities are everywhere in the real world! They're used in finance to set budgets and investment limits, in science to describe ranges of experimental results, in engineering to design structures that can withstand certain loads, and even in everyday situations like figuring out how much you can spend at the grocery store. Understanding inequalities is a valuable skill that can help you make informed decisions and solve problems in a wide range of contexts. It's like having a superpower that allows you to see the world in a more mathematical way.

As you continue your mathematical journey, remember that practice makes perfect. The more you work with inequalities and other algebraic concepts, the more comfortable and confident you'll become. Don't be afraid to make mistakes – they're part of the learning process. And most importantly, have fun! Math is a fascinating and powerful tool that can unlock a world of possibilities. So, keep exploring, keep questioning, and keep learning! You've got this!