Solving Compound Inequalities A Step-by-Step Guide

Hey guys! Let's dive into the world of compound inequalities and tackle a problem that might seem a bit tricky at first. We're going to break it down step by step, making sure you not only understand the solution but also the process behind it. So, grab your thinking caps, and let's get started!

Understanding Compound Inequalities

Before we jump into solving the specific inequality, let's chat a bit about what compound inequalities actually are. Think of them as two or more inequalities joined together by the words "and" or "or." The word "and" means that both inequalities must be true at the same time, while "or" means that at least one of the inequalities must be true. This distinction is crucial because it affects how we interpret and represent the solution. When dealing with compound inequalities connected by "and, " we are looking for the intersection of the solutions, the values that satisfy both inequalities simultaneously. On the other hand, when the inequalities are connected by "or," we seek the union of the solutions, any value that satisfies at least one of the inequalities. This fundamental difference dictates the approach we take to solve and represent the solution set.

For example, consider the difference between solving x > 3 and x < 5 (an "and" scenario) versus solving x > 3 or x < 5 (an "or" scenario). In the first case, we need values of x that are both greater than 3 and less than 5, which gives us a bounded interval. In the second case, we need values of x that are either greater than 3 or less than 5, which, in this instance, happens to include all real numbers because any number will satisfy at least one of the conditions. Understanding this difference is not just about getting the right answer; it's about grasping the underlying logic of inequalities and how they interact. The ability to discern between "and" and "or" conditions allows for a more nuanced problem-solving approach, enabling you to tackle more complex compound inequalities with confidence. This foundational knowledge is invaluable, serving as a cornerstone for more advanced mathematical concepts and applications. So, let’s keep this distinction in mind as we move forward to solve our inequality.

The Specific Compound Inequality: A Closer Look

Now, let's zoom in on the compound inequality we're going to solve: 4x + 2 ≤ -6 and 4x ≤ 0. Notice the "and"? That means we need to find the values of x that satisfy both inequalities. It's like a double requirement – x has to play by both sets of rules. Before we start crunching numbers, it's always a good idea to take a mental snapshot of what we're dealing with. We have two linear inequalities, and we're looking for the overlap in their solution sets. This "and" condition is key, as it narrows down our solution to only the values that make both statements true. If it were an "or," we'd be looking for values that satisfy either one, opening up our solution set considerably. This initial observation helps set the stage for our algebraic manipulations, ensuring we stay focused on the core task: finding the common ground between these two inequalities. Remember, the goal is not just to find a solution, but to understand why it's the solution. By recognizing the "and" condition upfront, we're setting ourselves up for a more intuitive and meaningful problem-solving experience. So, with this understanding firmly in place, let's roll up our sleeves and start solving each inequality individually.

Solving the First Inequality: 4x + 2 ≤ -6

Alright, let's tackle the first part of our compound inequality: 4x + 2 ≤ -6. To solve for x, we need to isolate it on one side of the inequality. Think of it like solving a regular equation, but with the added consideration that multiplying or dividing by a negative number flips the inequality sign. First, we'll subtract 2 from both sides to get 4x ≤ -8. This step helps us simplify the inequality by removing the constant term from the left side, bringing us closer to isolating x. It's a fundamental algebraic move that preserves the inequality, ensuring that our solution remains valid. Next, we'll divide both sides by 4. Since we're dividing by a positive number, the inequality sign stays the same. This gives us x ≤ -2. Boom! We've solved the first inequality. This result tells us that any value of x that is less than or equal to -2 will satisfy this particular inequality. But remember, we're dealing with a compound inequality, so this is just one piece of the puzzle. We still need to consider the second inequality and find the values of x that satisfy both. This individual solution serves as a crucial component in our quest for the overall solution set. So, with x ≤ -2 in our toolbox, let's move on to the second inequality and see what it has in store for us. Remember, staying organized and methodical is key to successfully navigating compound inequalities.

Solving the Second Inequality: 4x ≤ 0

Now, let's shift our focus to the second inequality: 4x ≤ 0. This one's a bit simpler, which is always a nice bonus! Again, our goal is to isolate x. To do this, we'll divide both sides of the inequality by 4. Since 4 is a positive number, we don't need to worry about flipping the inequality sign. This gives us x ≤ 0. And there you have it – the solution to our second inequality. This tells us that any value of x that is less than or equal to 0 will make this statement true. Now we have two pieces of the puzzle: x ≤ -2 from the first inequality and x ≤ 0 from the second. But remember, we're dealing with an "and" condition, so we need to find the values of x that satisfy both of these inequalities. It's like finding the overlapping region in a Venn diagram. The solution set must be a subset of both individual solutions. With both individual solutions in hand, we're now ready to bring them together and determine the solution to the entire compound inequality. This step involves careful consideration of the "and" condition and how it shapes the final answer. So, let's proceed with confidence and piece together the solution.

Finding the Solution Set: The Overlap

Okay, we've solved both inequalities individually. We know that x ≤ -2 and x ≤ 0. Now comes the crucial part: finding the solution set for the entire compound inequality. Remember, the "and" means we need to find the values of x that satisfy both inequalities simultaneously. Think of it as finding the overlap between the two solution sets. To visualize this, it can be helpful to imagine a number line. We have one solution set that includes all numbers less than or equal to -2, and another that includes all numbers less than or equal to 0. The overlap is the range of numbers that belong to both sets. If you picture this, you'll see that the numbers that are both less than or equal to -2 and less than or equal to 0 are simply the numbers less than or equal to -2. In other words, if a number is less than or equal to -2, it's automatically also less than or equal to 0. But the reverse isn't true – a number less than or equal to 0 might not be less than or equal to -2 (for example, -1). Therefore, the solution set for the compound inequality is x ≤ -2. This means that any value of x that is less than or equal to -2 will satisfy both original inequalities. We've successfully navigated the "and" condition and arrived at our final solution. But before we celebrate, let's make sure we can represent this solution in the correct format.

Expressing the Solution

So, we've determined that the solution set for the compound inequality is x ≤ -2. Now, let's talk about how to express this solution in the format you might see in a multiple-choice question or on a test. Often, you'll be asked to write the solution in interval notation or to select the correct graph representing the solution set. In interval notation, x ≤ -2 is written as (-∞, -2]. The parenthesis on the left indicates that negative infinity is not included (since it's not a specific number), and the square bracket on the right indicates that -2 is included in the solution set. This notation is a concise and standard way to represent a range of numbers. Graphically, the solution would be represented on a number line with a closed circle (or a square bracket) at -2 and a line extending to the left, indicating all numbers less than -2. The closed circle (or bracket) signifies that -2 is part of the solution. Understanding these different representations is crucial for effectively communicating your answer and for correctly interpreting solutions presented in various formats. It's not enough to just find the solution; you need to be able to express it clearly and accurately. This skill is invaluable in mathematics and beyond, as it allows you to convey complex information in a precise and understandable way. So, make sure you're comfortable with both interval notation and graphical representations of inequalities.

Final Answer and Conclusion

Wrapping things up, the solution set of the compound inequality 4x + 2 ≤ -6 and 4x ≤ 0 is x ≤ -2. In interval notation, this is expressed as (-∞, -2]. We arrived at this solution by first solving each inequality individually and then finding the intersection of their solution sets, guided by the "and" condition. Remember, solving compound inequalities involves a combination of algebraic manipulation and logical reasoning. It's not just about following steps; it's about understanding why those steps lead to the correct answer. By breaking down the problem into smaller parts, visualizing the solution sets, and paying close attention to the connecting words ("and" or "or"), you can confidently tackle a wide range of compound inequalities. So, keep practicing, and don't be afraid to ask questions. You've got this! Understanding compound inequalities is a fundamental skill in algebra, and mastering it will open doors to more advanced mathematical concepts. Keep up the great work, and happy solving!

A. The solution set of the compound inequality is (-∞, -2]