Solving Inequalities A Step-by-Step Guide
In the realm of mathematics, inequalities play a crucial role in defining relationships between values that are not necessarily equal. Unlike equations that establish equality, inequalities express a range of possible values that satisfy a given condition. This article delves into the intricacies of solving inequalities, providing a step-by-step guide to navigate these mathematical expressions effectively. We will explore the fundamental principles, techniques, and nuances involved in solving inequalities, equipping you with the tools to confidently tackle a wide array of problems. Understanding inequalities is not only essential for mathematical proficiency but also has practical applications in various fields, including economics, engineering, and computer science. This article will focus on solving a compound inequality, which involves two or more inequalities connected by the words "or" or "and." Compound inequalities require careful consideration of the individual inequalities and their relationship to each other to arrive at the correct solution set. By mastering the techniques presented here, you will gain a solid foundation for solving more complex mathematical problems and applying these skills to real-world scenarios.
Understanding Inequalities
Before diving into the process of solving inequalities, it's crucial to grasp the fundamental concepts and notation involved. Inequalities use specific symbols to compare values, each with its distinct meaning: less than (<), greater than (>), less than or equal to (≤), and greater than or equal to (≥). These symbols allow us to express relationships where one value is not precisely equal to another, but rather falls within a certain range. The solutions to inequalities are often represented graphically on a number line, providing a visual representation of the range of values that satisfy the inequality. This visual aid is particularly helpful in understanding the solution sets of compound inequalities, where multiple conditions must be met. To illustrate, consider the inequality x > 3. This inequality states that x can be any value greater than 3, but not including 3 itself. On a number line, this would be represented by an open circle at 3 (indicating that 3 is not included) and a line extending to the right, representing all values greater than 3. Similarly, the inequality x ≤ 5 includes all values less than or equal to 5, represented by a closed circle at 5 (indicating inclusion) and a line extending to the left. Understanding these basic concepts and notations is essential for effectively solving and interpreting inequalities. The ability to translate between symbolic notation and graphical representation is a key skill in mastering inequalities. Compound inequalities, which we will explore in detail, build upon these foundational concepts, requiring us to consider multiple inequalities and their combined solution sets.
Solving the Given Inequality
Now, let's tackle the specific inequality presented: (3/2)x + 3 < 2 or 3x + (1/5) ≥ 1. This is a compound inequality, meaning it consists of two separate inequalities connected by the word "or." To solve this, we need to solve each inequality individually and then combine the solutions based on the "or" condition. Let's start with the first inequality: (3/2)x + 3 < 2. Our goal is to isolate x on one side of the inequality. First, we subtract 3 from both sides: (3/2)x < 2 - 3, which simplifies to (3/2)x < -1. Next, to isolate x, we multiply both sides by the reciprocal of 3/2, which is 2/3: x < (-1) * (2/3). This gives us x < -2/3. So, the solution to the first inequality is all values of x less than -2/3. Now, let's move on to the second inequality: 3x + (1/5) ≥ 1. Again, we want to isolate x. First, subtract 1/5 from both sides: 3x ≥ 1 - (1/5), which simplifies to 3x ≥ 4/5. Next, divide both sides by 3 to isolate x: x ≥ (4/5) / 3. This gives us x ≥ 4/15. Thus, the solution to the second inequality is all values of x greater than or equal to 4/15. Since the original inequality uses the word "or," the final solution is the union of the solutions to the individual inequalities. This means that any value of x that satisfies either x < -2/3 or x ≥ 4/15 is a solution to the compound inequality. We can represent this solution set on a number line by shading the region to the left of -2/3 (not including -2/3) and the region to the right of 4/15 (including 4/15). Understanding how to solve inequalities individually and then combine their solutions based on the connecting word ("or" or "and") is crucial for mastering compound inequalities. In this case, the "or" condition means that the solution set includes all values that satisfy either inequality.
Expressing the Solution
Having solved the individual inequalities, we now need to express the complete solution to the compound inequality. We found that x < -2/3 or x ≥ 4/15. This means that any value of x that is either less than -2/3 or greater than or equal to 4/15 will satisfy the original inequality. There are several ways to represent this solution. We've already mentioned the number line representation, which provides a visual understanding of the solution set. We can also express the solution using interval notation, a concise way to represent ranges of values. In interval notation, we use parentheses to indicate that an endpoint is not included in the solution set and brackets to indicate that it is included. For the inequality x < -2/3, the interval notation is (-∞, -2/3). The parenthesis next to -∞ indicates that negative infinity is not a specific number and is never included, and the parenthesis next to -2/3 indicates that -2/3 itself is not included in the solution. For the inequality x ≥ 4/15, the interval notation is [4/15, ∞). The bracket next to 4/15 indicates that 4/15 is included in the solution, and the parenthesis next to ∞ indicates that infinity is not a specific number and is never included. Since the inequalities are connected by "or," we use the union symbol (∪) to combine the intervals. Therefore, the complete solution in interval notation is (-∞, -2/3) ∪ [4/15, ∞). This notation clearly and concisely represents all the values of x that satisfy the compound inequality. Understanding interval notation is essential for expressing solutions to inequalities in a standardized and easily interpretable format. It allows us to communicate the solution set precisely and avoid ambiguity.
Answering the Question
The original question presents the solution in a specific format: x < -[?]/□ or x ≥ □. Our task is to fill in the blanks based on the solution we derived. From our work, we found that x < -2/3 or x ≥ 4/15. Comparing this with the given format, we can directly fill in the blanks. For the first inequality, x < -[?]/□, we see that the numerator should be 2 and the denominator should be 3. So, the first blank should be filled with 2, and the box should contain 3. For the second inequality, x ≥ □, the value on the right side is 4/15. Therefore, the box should be filled with 4/15. So, the complete solution in the given format is x < -2/3 or x ≥ 4/15. This matches the solution we obtained through our step-by-step process. By carefully solving each inequality and then combining the solutions based on the "or" condition, we were able to accurately determine the values that satisfy the given compound inequality. This exercise demonstrates the importance of breaking down complex problems into smaller, manageable steps and systematically applying the appropriate techniques. Filling in the blanks in the given format reinforces our understanding of how the solution is represented and allows us to verify our answer.
Conclusion
Solving inequalities is a fundamental skill in mathematics with applications in various fields. In this article, we've provided a comprehensive guide to solving a compound inequality, walking through each step in detail. We started by understanding the basic concepts and notation of inequalities, then moved on to solving the individual inequalities within the compound inequality. We carefully isolated the variable in each inequality, paying attention to the direction of the inequality sign when multiplying or dividing by negative numbers. Once we had the solutions to the individual inequalities, we combined them based on the "or" condition, which means that any value satisfying either inequality is part of the solution set. We then discussed different ways to represent the solution, including graphically on a number line and using interval notation. Finally, we applied our solution to answer the specific question presented, filling in the blanks in the given format. By mastering these techniques, you'll be well-equipped to solve a wide range of inequalities and apply these skills to more complex mathematical problems. Remember to practice regularly and break down problems into smaller steps to ensure accuracy and understanding. The ability to solve inequalities is not just a mathematical skill; it's a valuable tool for problem-solving and critical thinking in many areas of life. Whether you're analyzing data, making decisions, or simply trying to understand the world around you, the principles of inequalities can help you make informed choices and arrive at sound conclusions.
