Hey guys! Today, we're diving into the fascinating world of compound inequalities. You know, those mathematical expressions that combine two or more inequalities using "or" or "and"? Specifically, we're going to tackle the compound inequality 33.2 ≤ 2x - 3.4 or 4.7 - 6x ≥ -18.1. Buckle up, because by the end of this guide, you'll be solving these like a pro! Understanding the process of solving inequalities is essential not only for academic success but also for real-world problem-solving scenarios where constraints and limitations play a crucial role. Grasping the concepts behind compound inequalities enhances your analytical skills, allowing you to make informed decisions in various fields such as economics, engineering, and computer science. So, let's embark on this mathematical journey and empower ourselves with the ability to navigate the complexities of inequalities.
Understanding Compound Inequalities
Before we jump into the solution, let's make sure we're all on the same page about what compound inequalities are. Think of them as two separate inequalities hanging out together, connected by either "or" or "and." The word "or" means that a solution needs to satisfy at least one of the inequalities, while "and" means it needs to satisfy both.
The key to mastering compound inequalities lies in understanding the logical operators that connect them. The word "or" implies a union of solutions, meaning any value that satisfies either inequality is part of the solution set. On the other hand, "and" signifies an intersection, where only values that satisfy both inequalities simultaneously are considered solutions. This distinction is crucial when interpreting and representing the solution set. For instance, in our example, the presence of "or" suggests that the solution set will include values that satisfy either 33.2 ≤ 2x - 3.4 or 4.7 - 6x ≥ -18.1, potentially resulting in a broader range of solutions compared to a compound inequality connected by "and". Understanding this fundamental concept will pave the way for a seamless solution process.
Breaking Down the Problem
Our compound inequality is 33.2 ≤ 2x - 3.4 or 4.7 - 6x ≥ -18.1. See? Two inequalities joined by "or." Our mission is to find all the values of x that make this statement true. To do that, we will solve each inequality separately and then combine their solutions based on the "or" condition. Remember, the or condition means that if x satisfies either of the inequalities, it is part of the solution set. This is a crucial point because it dictates how we interpret and combine the individual solutions. Solving each inequality independently allows us to isolate x and determine the range of values that satisfy each condition. Once we have these individual solution sets, we can then merge them using the rules of logical or, which essentially means taking the union of the two sets. This approach simplifies the problem by breaking it down into manageable steps, ensuring that we capture all possible solutions.
Solving the First Inequality: 33.2 ≤ 2x - 3.4
Let's tackle the first inequality: 33.2 ≤ 2x - 3.4. To isolate x, we'll follow the standard algebraic steps. First, we'll add 3.4 to both sides of the inequality. This will help us get rid of the constant term on the right side, bringing us closer to isolating x. Remember, whatever operation we perform on one side of the inequality, we must also perform on the other side to maintain the balance. This is a fundamental principle in solving inequalities and equations alike. By adding 3.4 to both sides, we simplify the inequality and make it easier to work with. This step is crucial because it sets the stage for the subsequent steps, ultimately leading us to the solution for x. The goal here is to manipulate the inequality in such a way that x is by itself on one side, revealing the range of values that satisfy the inequality.
Step-by-Step Solution
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Add 3.4 to both sides: 33. 2 + 3.4 ≤ 2x - 3.4 + 3.4 34. 6 ≤ 2x
Adding 3.4 to both sides cancels out the -3.4 on the right, leaving us with 36.6 ≤ 2x. This is a simplified form of the original inequality, making it easier to proceed with isolating x. The next step involves dividing both sides by 2, which will finally give us the solution for x. Each step in solving an inequality is designed to gradually isolate the variable, allowing us to determine the range of values that satisfy the given condition. This methodical approach ensures accuracy and clarity in the solution process.
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Divide both sides by 2: 35. 6 / 2 ≤ 2x / 2 36. 3 ≤ x
Dividing both sides by 2 isolates x, giving us 18.3 ≤ x. This means that x must be greater than or equal to 18.3 for the first inequality to hold true. We've successfully solved the first part of our compound inequality! But remember, we're not done yet. We still need to solve the second inequality and then combine the solutions. This result tells us that any value of x that is 18.3 or greater will satisfy the first part of our compound inequality. However, to find the complete solution, we must now turn our attention to the second inequality and see how its solutions interact with this result.
Solution Set for the First Inequality
The solution to the first inequality, 33.2 ≤ 2x - 3.4, is x ≥ 18.3. This means any number greater than or equal to 18.3 will satisfy this part of the compound inequality. Think of it as a number line where everything from 18.3 to positive infinity is shaded in. This is a crucial piece of the puzzle, but it's not the whole picture. We still have the second inequality to consider. Visualizing the solution set on a number line is a helpful way to understand the range of values that satisfy the inequality. It allows us to see at a glance which numbers are included in the solution and which are not. This visual representation will be particularly useful when we combine the solutions of both inequalities, as it will help us determine the overall solution set of the compound inequality.
Solving the Second Inequality: 4.7 - 6x ≥ -18.1
Now, let's move on to the second inequality: 4.7 - 6x ≥ -18.1. This one looks a little different, but the same principles apply. Our goal is still to isolate x. We'll start by subtracting 4.7 from both sides to get the term with x by itself. This step is analogous to what we did in the first inequality, where we aimed to isolate the variable term. Subtracting 4.7 from both sides simplifies the inequality and moves us closer to isolating x. Remember, maintaining balance is key, so we perform the same operation on both sides. This process of isolating x is fundamental to solving inequalities and equations, and it requires a careful application of algebraic principles.
Step-by-Step Solution
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Subtract 4.7 from both sides: 4. 7 - 6x - 4.7 ≥ -18.1 - 4.7 -6x ≥ -22.8
Subtracting 4.7 from both sides gives us -6x ≥ -22.8. Notice the negative sign in front of the x term – that's a key indicator that we'll need to be careful in the next step. When we divide by a negative number, we'll need to flip the inequality sign. This is a crucial rule to remember when working with inequalities. Neglecting to flip the sign would lead to an incorrect solution set. This step highlights the importance of paying close attention to the details when solving inequalities.
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Divide both sides by -6 (and flip the inequality sign!): -6x / -6 ≤ -22.8 / -6 x ≤ 3.8
Dividing both sides by -6 gives us x ≤ 3.8. Remember, we flipped the inequality sign because we divided by a negative number! This means that x must be less than or equal to 3.8 for the second inequality to hold true. This step is a critical point in the solution process. The rule of flipping the inequality sign when dividing or multiplying by a negative number is a fundamental concept in algebra. Understanding and applying this rule correctly is essential for obtaining the correct solution set. The result x ≤ 3.8 tells us that any value of x less than or equal to 3.8 will satisfy the second inequality.
Solution Set for the Second Inequality
The solution to the second inequality, 4.7 - 6x ≥ -18.1, is x ≤ 3.8. This means any number less than or equal to 3.8 will satisfy this part of the compound inequality. Again, we can picture this on a number line, where everything from negative infinity up to 3.8 is shaded in. We now have the solution sets for both inequalities, but we're not quite finished. The next step is to combine these solutions using the or condition. This involves understanding how the or operator affects the overall solution set and how it relates to the individual solutions we've found.
Combining the Solutions with "Or"
Okay, we've solved both inequalities separately. Now comes the fun part: combining the solutions! Remember, our compound inequality uses "or," which means we want all values of x that satisfy either x ≥ 18.3 or x ≤ 3.8. In mathematical terms, we're looking for the union of the two solution sets. This means we'll include any value that satisfies at least one of the inequalities. Understanding the concept of union is crucial here. It's like merging two sets of numbers together, keeping all the unique elements from both sets. In the context of inequalities, it means that our final solution set will encompass all values that make either inequality true.
Visualizing the Solution Sets
Think of two number lines: One shaded from 18.3 to the right (for x ≥ 18.3), and the other shaded from 3.8 to the left (for x ≤ 3.8). Since we have "or," we take everything that's shaded. This visualization really helps to see the final solution. It provides a clear picture of the range of values that satisfy the compound inequality. The shaded regions on the number lines represent the solution sets of the individual inequalities, and the combined shaded region represents the solution set of the compound inequality. This visual approach simplifies the process of understanding and interpreting the solution.
The Final Solution
The solution to the compound inequality 33.2 ≤ 2x - 3.4 or 4.7 - 6x ≥ -18.1 is x ≤ 3.8 or x ≥ 18.3. That's it! We've found all the values of x that make the original statement true. This means any number less than or equal to 3.8, or any number greater than or equal to 18.3, is a solution. This result may seem a bit complex, but it accurately represents the solution set of the compound inequality. It's a range of values that extends from negative infinity up to 3.8, and then picks up again from 18.3 to positive infinity. This type of solution is common with or compound inequalities, where the solution set can be disjointed.
Representing the Solution
There are a few ways we can represent this solution. We've already used inequality notation (x ≤ 3.8 or x ≥ 18.3). We can also use interval notation or graph it on a number line.
Interval Notation
In interval notation, the solution is written as (-∞, 3.8] ∪ [18.3, ∞). The parentheses indicate that the endpoint is not included, while the square brackets indicate that it is. The