Solve Inequality: Find Interval For X
Hey guys! Let's dive into this math problem together and figure out how to solve this inequality. We're going to break it down step by step, so don't worry if it looks a little intimidating at first. We need to find the interval that includes all possible values of x in the inequality $-3(6-2x) \geq 4x+12$. Inequalities, at first glance, might seem a bit like equations, but instead of a simple equals sign, we're dealing with greater than or equal to, less than, or other similar symbols. This means our solution won't be a single number, but a range of numbers. This range is what we call an interval, and it represents all the values that x can take while still making the inequality true. So, buckle up, and let's get started on the journey to finding the right interval!
Understanding the Problem
Before we jump into the calculations, let’s make sure we really understand the inequality. Remember, we're looking for all the x values that make the statement $-3(6-2x) \geq 4x+12$ true. This means we need to do some algebraic manipulation to isolate x on one side of the inequality. Think of it like a balancing act – whatever we do to one side, we also need to do to the other to keep things fair. Understanding the underlying principles of inequalities is crucial. We're not just plugging and chugging numbers; we're exploring the very fabric of mathematical relationships. For instance, multiplying or dividing both sides by a negative number flips the inequality sign – a small but vital detail.
The first step in solving any inequality is to simplify both sides as much as possible. This often involves distributing any terms outside parentheses, combining like terms, and generally cleaning things up. A clean, simplified inequality is much easier to work with and reduces the chances of making errors along the way. So, let's focus on this initial simplification process and see how we can transform our given inequality into a more manageable form. This preparatory stage is like setting the foundation for a building – a solid foundation ensures a sturdy and accurate solution.
Step-by-Step Solution
Okay, let's break down the solution step-by-step. First, we need to distribute the -3 on the left side of the inequality:
This becomes:
See? We're making progress already! Next, we want to get all the x terms on one side and the constants on the other. Let's subtract 4x from both sides:
This simplifies to:
Now, let's add 18 to both sides to isolate the x term further:
Which gives us:
Finally, to solve for x, we divide both sides by 2:
And there we have it:
This is the key result! This inequality tells us that x can be any value greater than or equal to 15.
Interpreting the Solution
So, what does $x \geq 15$ actually mean? Well, it means that any number 15 or larger will satisfy the original inequality. Think of it on a number line – we're starting at 15 and shading everything to the right. This “shading” represents the infinite number of solutions that fit our inequality. Understanding the graphical representation of inequalities can provide a powerful visual aid, especially when dealing with more complex scenarios. For instance, when working with systems of inequalities, the overlapping shaded regions represent the solution set that satisfies all inequalities simultaneously.
To express this as an interval, we use the notation $[15, \infty)$. The square bracket '[' indicates that 15 is included in the solution (because of the “equal to” part of the $ \geq $ sign), and the parenthesis ')' next to infinity means that infinity is not a number, but rather a concept indicating that the interval continues indefinitely in the positive direction. Mastering interval notation is essential for communicating solutions clearly and concisely in mathematics. It's a universal language that allows mathematicians and students alike to understand the precise range of values that satisfy a given condition. So, let's make sure we're comfortable with both the inequality notation and its corresponding interval representation.
Identifying the Correct Interval
Now, let's go back to the answer choices provided in the problem. We need to find the interval that matches our solution, which is $x \geq 15$.
- A. $(-\infty, -3]$: This interval includes all numbers less than or equal to -3. This is not what we found.
- B. $[-3, \infty)$: This interval includes all numbers greater than or equal to -3. This is also not our solution.
- C. $(-\infty, 15]$: This interval includes all numbers less than or equal to 15. Close, but we need greater than or equal to 15.
- D. $[15, \infty)$: This is it! This interval includes all numbers greater than or equal to 15, which perfectly matches our solution $x \geq 15$.
So, the correct answer is D. $[15, \infty)$.
Choosing the correct answer from a set of options is a critical skill in mathematics. It's not enough to simply solve the problem; you must also be able to translate your solution into the format required by the question. This often involves careful consideration of interval notation, inequality symbols, and the specific context of the problem.
Common Mistakes to Avoid
Let's chat about some common mistakes people make when solving inequalities. One biggie is forgetting to flip the inequality sign when multiplying or dividing by a negative number. Remember that rule! It’s like a little mathematical speed bump that can throw you off course if you’re not careful. Being mindful of these common pitfalls can significantly improve your accuracy and confidence when tackling inequality problems. It's not just about knowing the rules, but also about understanding why they exist and how they apply in different situations. Think of it as developing a mathematical intuition – a sense for what's right and wrong based on the underlying principles.
Another mistake is messing up the order of operations. Always follow PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Getting the order wrong can lead to incorrect simplification and ultimately, the wrong answer. Also, be careful when distributing negative signs. It's easy to make a small error there, which can throw off the entire solution. Practice is key to avoiding these mistakes. The more you work with inequalities, the more comfortable you'll become with the process and the less likely you are to fall into these common traps.
Finally, always double-check your answer. Plug a value from your solution interval back into the original inequality to make sure it works. This is a great way to catch any errors you might have made along the way. Verification is a crucial step in the problem-solving process. It's not enough to simply arrive at an answer; you must also ensure that your answer is correct and that it makes sense in the context of the problem.
Real-World Applications
So, why are inequalities important anyway? Well, they pop up all over the place in real life! Think about things like budgets (you can't spend more than you have), speed limits (you can't go faster than the limit), or even recipe measurements (you need at least a certain amount of ingredients). Exploring these real-world applications can make the study of inequalities much more engaging and meaningful. It's not just abstract mathematics; it's a powerful tool for understanding and navigating the world around us.
Inequalities are also fundamental in fields like economics, where they're used to model supply and demand, and in engineering, where they're used to design structures that can withstand certain loads. Even in computer science, inequalities play a role in optimization algorithms and resource allocation. So, the skills you develop in solving inequalities are not just for the classroom; they're valuable assets in a wide range of professional fields. Understanding the practical applications of mathematical concepts can inspire a deeper appreciation for the subject and motivate you to excel in your studies.
Conclusion
We did it! We successfully solved the inequality and found the interval that includes all possible values of x. Remember, the key is to simplify the inequality step-by-step, isolate the variable, and then interpret the solution in terms of an interval. Keep practicing, and you'll become a master of inequalities in no time! Mastering inequalities is a significant step in your mathematical journey. It's a foundational concept that will serve you well in more advanced topics, such as calculus and linear algebra. So, embrace the challenge, persevere through the difficulties, and celebrate your successes along the way!
If you found this helpful, give it a thumbs up, and let me know what other math problems you'd like to tackle next time. Keep learning, guys! Remember that math is like building blocks, each concept building upon the previous. So, a strong foundation in inequalities will make your future math endeavors much smoother and more enjoyable.
