Solving Inequalities Finding The Interval For -3(6-2x) >= 4x + 12
Hey everyone! Today, we're diving into the world of inequalities, specifically tackling the problem of finding the interval that encompasses all possible values of x for the inequality $-3(6-2x) ">=" 4x + 12$. This is a classic algebra problem, and by breaking it down step by step, we'll not only arrive at the correct answer but also solidify our understanding of how to solve such inequalities. So, grab your pencils, and let's get started!
Decoding the Inequality: A Step-by-Step Approach
When faced with an inequality like this, the key is to simplify it systematically. Our goal is to isolate x on one side of the inequality sign. Let's walk through the process:
1. Distribute the -3: The first step involves distributing the -3 across the terms inside the parentheses. This means multiplying -3 by both 6 and -2x. Remember, a negative times a negative results in a positive! So, we have:
This step is crucial for removing the parentheses and bringing all the x terms into play. By carefully distributing, we've laid the groundwork for the next steps.
2. Gather the x terms: Now, let's bring all the x terms to one side of the inequality. To do this, we can subtract 4x from both sides. This maintains the balance of the inequality while grouping the x terms together:
Notice how subtracting 4x from both sides cancels out the 4x on the right side, leaving us with just the constant term.
3. Isolate the x term: Next, we want to isolate the term with x (which is 2x in this case). To do this, we'll add 18 to both sides of the inequality. This will cancel out the -18 on the left side:
Adding 18 to both sides keeps the inequality balanced and brings us closer to isolating x.
4. Solve for x: Finally, to get x by itself, we need to divide both sides of the inequality by 2:
And there we have it! We've successfully isolated x and found that x is greater than or equal to 15. This is a critical result, so make sure you understand how we got here.
Interpreting the Solution: Interval Notation
Now that we've solved for x, we need to express our solution in interval notation. Remember, interval notation is a way of representing a set of numbers using brackets and parentheses. Since our solution is x ">=" 15, this means that x can be 15 or any number greater than 15.
In interval notation, we represent this as [15, ∞). Let's break down what this means:
- [15: The square bracket indicates that 15 is included in the solution set. This is because our inequality is "greater than or equal to" 15.
- , ∞): The infinity symbol (∞) represents positive infinity, meaning the solution continues indefinitely in the positive direction. The parenthesis next to the infinity symbol indicates that infinity is not included in the solution set (because infinity is not a specific number).
Therefore, the interval [15, ∞) represents all numbers from 15 (inclusive) up to positive infinity. This is the interval that contains all possible values of x that satisfy our original inequality.
Connecting the Dots: The Correct Answer
Looking back at the options provided, we can now confidently identify the correct answer:
A. $(-\infty, -3]$ B. $[-3, \infty)$ C. $(-\infty, 15]$ D. $[15, \infty)$
Our solution, [15, ∞), matches option D. So, the correct answer is D. [15, ∞). Great job, guys! You've successfully navigated through this inequality problem.
Why Other Options are Incorrect
To truly master this concept, let's briefly discuss why the other options are incorrect. This will further solidify our understanding of inequalities and interval notation.
- A. (-∞, -3]: This interval represents all numbers less than or equal to -3. This is completely different from our solution, which includes numbers greater than or equal to 15.
- B. [-3, ∞): This interval represents all numbers greater than or equal to -3. While it includes some positive numbers, it doesn't start at 15, which is the lower bound of our solution.
- C. (-∞, 15]: This interval represents all numbers less than or equal to 15. It includes numbers up to 15, but it doesn't include any numbers greater than 15, which are part of our solution.
By understanding why these options are incorrect, we reinforce our understanding of the correct solution and the nuances of interval notation.
Practice Makes Perfect: Tackling Similar Inequalities
Now that we've conquered this inequality, let's talk about how to tackle similar problems. The key is to apply the same systematic approach we used here:
- Simplify: Distribute, combine like terms, and get rid of any parentheses.
- Isolate the variable: Use addition and subtraction to get all the x terms on one side and the constant terms on the other.
- Solve for the variable: Divide (or multiply, if necessary) to get x by itself.
- Express the solution in interval notation: Use brackets and parentheses to represent the set of solutions.
Remember, practice is crucial! The more you work with inequalities, the more comfortable you'll become with the process. Try solving different types of inequalities, including those with fractions, decimals, and compound inequalities.
The Power of Inequalities: Real-World Applications
Inequalities aren't just abstract mathematical concepts; they have numerous real-world applications. Think about situations where you need to represent a range of values, such as:
- Budgeting: You might have a budget constraint, meaning you can spend up to a certain amount of money. This can be represented using an inequality.
- Speed limits: Speed limits on roads are expressed as maximum speeds, which can be represented using an inequality.
- Temperature ranges: A recipe might require you to bake something at a temperature between two values, which can be represented using a compound inequality.
- Eligibility criteria: Many programs have eligibility criteria based on age, income, or other factors. These criteria are often expressed using inequalities.
Understanding inequalities allows us to model and solve these real-world problems. By mastering the techniques we've discussed today, you'll be well-equipped to tackle a wide range of challenges.
Conclusion: Mastering Inequalities for Mathematical Success
Solving inequalities is a fundamental skill in algebra, and it's essential for success in higher-level math courses. By understanding the steps involved in simplifying, isolating, and solving inequalities, and by practicing regularly, you can build a strong foundation in this area. Remember to always express your solutions in interval notation, and don't be afraid to tackle challenging problems. With dedication and perseverance, you'll become a master of inequalities!
So there you have it, guys! We've successfully solved the inequality $-3(6-2x) "geq" 4x + 12$ and identified the correct interval as [15, ∞). Keep practicing, and you'll become an inequality-solving pro in no time! Thanks for joining me on this mathematical adventure.
