Unlocking Inequalities: Solving $3x-6>21$ Step-by-Step

Hey there, math enthusiasts! Today, we're diving headfirst into the world of inequalities, and we're going to solve the inequality $3x - 6 > 21$. Don't worry, it's not as scary as it sounds! In fact, with a few simple steps, we can crack this problem and understand the relationship between variables and values. Let's break it down and find out how to get to the answer, and more importantly, why the answer is what it is. This is a fundamental concept in algebra, and mastering it will set you up for success in more complex mathematical problems down the road. So, grab your pencils, and let's get started!

Understanding the Basics of Inequalities

Before we jump into the solution, let's quickly recap what an inequality is. Unlike equations, which use an equals sign (=), inequalities use symbols like 'greater than' (>), 'less than' (<), 'greater than or equal to' (≥), and 'less than or equal to' (≤). These symbols show a relationship between two values, indicating that one is larger or smaller than the other. In our case, $3x - 6 > 21$ means that the expression $3x - 6$ is greater than 21. Our goal is to isolate the variable x and find the range of values that make this inequality true. This is the heart of solving inequalities: determining the set of values that satisfy the given condition. Think of it like a puzzle where we must find the right pieces (values of x) to make the inequality balance correctly. When we solve an inequality, we're not just looking for a single answer, but rather a range of possible answers. This range is the set of all values of x for which the inequality holds true. Keep this in mind as we work through the steps to solve the given inequality. The power of understanding inequalities lies in their applications. They are critical in modeling real-world problems. For instance, determining the constraints for a budget, or finding the range of values for which a product is profitable. Inequalities help us describe these types of real-world scenarios. By practicing solving inequalities like $3x - 6 > 21$, you build a solid foundation. This knowledge is essential for tackling more advanced mathematical concepts and for making informed decisions in everyday situations.

Step-by-Step Solution: Unveiling the Answer

Alright, guys, let's get our hands dirty and solve this inequality, step by step! Remember, we want to isolate x on one side of the inequality. Here’s how we do it:

1. Isolate the term with x: Our first step is to get rid of that -6. To do that, we add 6 to both sides of the inequality. This keeps the inequality balanced. So, the inequality becomes: $3x - 6 + 6 > 21 + 6$. This simplifies to $3x > 27$.

2. Isolate x: Now, we need to get x all by itself. Since x is being multiplied by 3, we divide both sides of the inequality by 3. This gives us: $(3x)/3 > 27/3$.

3. Simplify: Finally, simplify both sides to find the value of x. This results in: $x > 9$.

And there you have it! The solution to the inequality $3x - 6 > 21$ is $x > 9$. This means any value of x that is greater than 9 will satisfy the original inequality. In simpler terms, if you plug in any number greater than 9 into the original expression $3x - 6$, the result will always be greater than 21. Keep in mind that when we solve inequalities, we're looking for the range of values of the variable that makes the statement true, not just a single value. This understanding of ranges is crucial in various applications of mathematics and is the essence of how we interpret solutions to inequality problems.

Decoding the Solution: What Does It Mean?

So, we found that $x > 9$. But what does this really mean? Well, it means that any number larger than 9 will make the original inequality true. Let's try some examples:

• If $x = 10$, then $3(10) - 6 = 30 - 6 = 24$. And, hey, 24 is greater than 21! Success!
• If $x = 11$, then $3(11) - 6 = 33 - 6 = 27$. Again, 27 is greater than 21. We're on a roll!
• If $x = 8$, then $3(8) - 6 = 24 - 6 = 18$. Nope, 18 is not greater than 21. That's why 8 is not a solution. The solution of $x > 9$ means that when we substitute $x$ with any value that is bigger than 9, the original inequality holds. This helps to visualize the set of numbers that solve our inequality. Understanding the solution allows you to apply it in different contexts and helps you verify the solution. The ability to verify the answer is an important skill when you want to solve an inequality.

Choosing the Correct Answer: The Final Verdict

Now, let's circle back to the multiple-choice options and see which one matches our solution: $x > 9$. Looking at the options, we see that (D) $x > 9$ is the correct answer. This is where your understanding of the steps and the solution pays off. You've successfully navigated the problem and arrived at the correct answer. This exercise is not just about finding the right choice, it's about building a solid foundation in understanding mathematical concepts, which is far more rewarding. Remember, solving inequalities is a fundamental skill in algebra and is essential for more advanced math topics. Being able to solve it correctly will give you a great advantage, so keep practicing!

Tips for Success: Mastering Inequalities

Here are a few handy tips to help you become a whiz at solving inequalities:

• Practice, practice, practice: The more you solve inequalities, the more comfortable you'll become. Work through a variety of problems to understand the different types and how to solve them.
• Understand the rules: Remember the key rules. When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. This is a common mistake, so be careful!
• Check your work: Always check your answer by plugging a value (or several) into the original inequality to make sure it works.
• Visualize: Use number lines to visualize the solution. This can help you understand the range of values that satisfy the inequality.

Beyond the Basics: Expanding Your Knowledge

Once you've mastered the basics, you can explore more advanced topics related to inequalities:

• Compound inequalities: These involve two or more inequalities joined together, such as $2 < x < 5$.
• Absolute value inequalities: These involve the absolute value of a variable, such as $|x| < 3$.
• Graphing inequalities: You can graph inequalities on a number line or coordinate plane to represent the solution visually.

By expanding your knowledge, you build a solid foundation. This knowledge will set you on the path to success in various areas of mathematics and its real-world applications. By consistently practicing and deepening your knowledge, you're not just solving equations, you're building a powerful skill set that extends far beyond the classroom.

Conclusion: Your Journey into Inequalities

Great job, guys! You've successfully solved the inequality $3x - 6 > 21$. You now understand the basic steps involved and, more importantly, why those steps work. Remember, the key is practice and understanding. Keep working at it, and you'll become a master of inequalities in no time! Keep exploring, keep questioning, and keep having fun with math! Happy solving, and see you next time!