Solving Inequalities Finding The Solution Set For -1.5(4x + 1) ≥ 4.5 - 2.5(x + 1)
Hey guys! Let's dive into a math problem together. Today, we're tackling an inequality question that might seem a bit tricky at first, but don't worry, we'll break it down step-by-step. Our main goal is to find the solution set for the inequality . Inequalities, just like equations, help us describe relationships between quantities, but instead of an equals sign, we use symbols like greater than or equal to (\\≥), less than or equal to (\\≤), greater than (>), or less than (<). Understanding how to solve inequalities is super important, especially when you're dealing with real-world problems where things aren't always exact. We need to consider a range of possible values. So, let's put on our math hats and get started!
Understanding the Problem
Before we jump into the calculations, it's crucial to understand the problem at hand. We have the inequality , and our mission is to find all the values of 'x' that make this statement true. In simpler terms, we're looking for a range of numbers that, when plugged in for 'x', will satisfy the inequality. This range of numbers is what we call the solution set. Inequalities are used everywhere, from figuring out how many items you can buy within a budget to determining the range of acceptable temperatures in a science experiment. Mastering these skills opens doors to many applications. The solution set can be expressed in a few different ways, and the multiple-choice answers give us a hint at the formats we might encounter such as an inequality (like x ≥ -1), or interval notation (like (-∞, -1]). So, let's keep these options in mind as we work through the problem.
Step-by-Step Solution
Okay, let's get our hands dirty and solve this inequality step by step. Solving inequalities is a lot like solving equations, but there's one key difference we need to remember: if we multiply or divide both sides by a negative number, we need to flip the direction of the inequality sign. Keep this golden rule in mind, and we'll be golden! First up, we'll distribute the numbers outside the parentheses on both sides of the inequality. This means multiplying -1.5 by both 4x and 1 on the left side, and -2.5 by both x and 1 on the right side. This gives us: -1.5 * 4x + (-1.5 * 1) ≥ 4.5 - (2.5 * x) - (2.5 * 1) which simplifies to -6x - 1.5 ≥ 4.5 - 2.5x - 2.5. Distributing correctly is super important because a small mistake here can throw off the whole solution. Next, we'll simplify both sides by combining any like terms. On the left side, we just have -6x - 1.5 for now. On the right side, we have 4.5 and -2.5, which we can combine. This gives us: -6x - 1.5 ≥ 2 - 2.5x. Now, let's move all the 'x' terms to one side of the inequality and the constants to the other side. We can do this by adding 2.5x to both sides and adding 1.5 to both sides. This gives us: -6x + 2.5x ≥ 2 + 1.5 which simplifies to -3.5x ≥ 3.5. We're almost there! To isolate 'x', we need to divide both sides by -3.5. But remember our golden rule! Since we're dividing by a negative number, we need to flip the inequality sign. This gives us: x ≤ 3.5 / -3.5 which simplifies to x ≤ -1. Hooray! We've found our solution.
Interpreting the Solution
Now that we've crunched the numbers, let's interpret what our solution x ≤ -1 actually means. This inequality tells us that any value of 'x' that is less than or equal to -1 will satisfy the original inequality. Think about it like a number line: our solution includes -1 and all the numbers stretching infinitely to the left. So, -2, -3, -10, -100, and so on, all work. But what about -0.5? Nope, that's greater than -1, so it's not part of our solution set. Understanding the solution isn't just about getting the right number, it's about grasping the concept and what it represents in the context of the problem. This understanding helps us apply these skills to real-world scenarios, too. Remember that we can represent this solution set in different ways. We already have it as an inequality (x ≤ -1), but we can also express it using interval notation.
Expressing the Solution Set
Let's talk about how to express the solution set. We've already got it in inequality form (x ≤ -1), which is super clear, but there's another way to write it that's commonly used in math: interval notation. Interval notation uses parentheses and brackets to show the range of values that are included in the solution. A parenthesis '(' or ')' means that the endpoint is not included, while a bracket '[' or ']' means that the endpoint is included. Since our solution is x ≤ -1, we want to include -1 in our solution set, and we want all the numbers less than -1, stretching all the way to negative infinity. Infinity is always represented with a parenthesis because we can't actually reach infinity, so we can't include it as an endpoint. Therefore, in interval notation, our solution set is (-∞, -1]. The negative infinity symbol (-∞) indicates that the interval extends indefinitely in the negative direction, and the bracket ']' next to -1 indicates that -1 is included in the solution set. Interval notation is a concise and standard way to represent solution sets, and it's good to be familiar with it. So, let's recap what we've done so far: we've solved the inequality, interpreted the solution, and now we know how to express it in both inequality and interval notation. We're doing great!
Analyzing the Answer Choices
Now, let's analyze the answer choices given in the problem. This is a crucial step because it helps us confirm our solution and make sure we haven't made any sneaky errors along the way. The answer choices are:
- A. x ≥ -1
- B. x ≥ 7/16
- C. (-∞, -1]
- D. (-∞, 7/16]
We found that the solution to the inequality is x ≤ -1. Looking at the answer choices, we can immediately eliminate options A and B because they represent solutions where x is greater than or equal to a certain value, not less than or equal to. Option C, (-∞, -1], is the interval notation for x ≤ -1, which matches our solution perfectly. Option D, (-∞, 7/16], represents all numbers less than or equal to 7/16, which is not our solution. So, by carefully working through the steps and comparing our answer to the choices, we can confidently select the correct answer. Always double-check your work and make sure the answer makes sense in the context of the original problem. This will help you avoid common mistakes and build your confidence in solving inequalities.
Final Answer
Alright guys, we've reached the end of our mathematical journey! After carefully solving the inequality and analyzing the answer choices, we've confidently determined that the correct answer is C. (-∞, -1]. This interval notation represents all values of x that are less than or equal to -1, which is the solution set to our inequality. We started by distributing, simplifying, and isolating 'x', remembering the crucial rule about flipping the inequality sign when dividing by a negative number. Then, we interpreted our solution and expressed it in interval notation. Finally, we matched our solution to the answer choices, confirming our result. Solving inequalities is a fundamental skill in mathematics, and the more you practice, the better you'll become. Remember to break down the problem into smaller steps, pay attention to the details, and always double-check your work. You got this!