Solving Inequalities Finding Values For S In 12s-20 ≤ 50-3s-25
In the realm of mathematics, inequalities play a crucial role in defining the range of possible solutions for a given problem. This article delves into the process of solving the inequality 12s-20 ≤ 50-3s-25, providing a step-by-step guide to determine the possible values for s. Understanding inequalities is fundamental in various fields, including algebra, calculus, and real-world applications. Our discussion will not only focus on the solution but also on the underlying principles that govern inequality manipulation. The ability to solve inequalities is not merely an academic exercise; it is a practical skill that can be applied to numerous real-life scenarios, such as budgeting, resource allocation, and decision-making processes. This article aims to equip you with the knowledge and skills necessary to confidently tackle inequalities and interpret their solutions. Let's embark on this mathematical journey together, unraveling the intricacies of inequalities and mastering the techniques to solve them effectively.
Understanding Inequalities
Before diving into the solution, let's understand the concept of inequalities. Inequalities are mathematical expressions that compare two values using symbols such as less than (<), greater than (>), less than or equal to (≤), and greater than or equal to (≥). Unlike equations that have definite solutions, inequalities often have a range of solutions. In the context of 12s-20 ≤ 50-3s-25, we aim to find all values of s that satisfy this condition. Grasping this fundamental concept is crucial for correctly interpreting and applying the solution. The distinction between equations and inequalities lies in the nature of their solutions. Equations typically have a finite set of solutions, whereas inequalities can have an infinite range of values that satisfy the condition. This range is often represented graphically on a number line or in interval notation. Furthermore, understanding the properties of inequalities is essential for manipulating them correctly. Operations such as adding or subtracting the same value from both sides, or multiplying or dividing by a positive number, preserve the inequality. However, multiplying or dividing by a negative number reverses the direction of the inequality, a critical point to remember when solving inequalities. The interplay between these concepts forms the bedrock of inequality solutions, enabling us to navigate the problem-solving process with clarity and precision.
Step-by-Step Solution
1. Simplify the Inequality
The first step involves simplifying both sides of the inequality. Combine like terms on the right side of the inequality: 50-3s-25 simplifies to 25-3s. Now, the inequality becomes 12s-20 ≤ 25-3s. This initial simplification is crucial for making the inequality more manageable and easier to solve. By combining the constant terms, we reduce the complexity of the expression, allowing us to focus on isolating the variable s. Simplification is a fundamental technique in algebra, enabling us to transform complex expressions into simpler, more understandable forms. This step not only streamlines the solution process but also reduces the likelihood of errors. In the context of inequalities, simplification often involves combining like terms, distributing coefficients, and removing parentheses. The goal is to create an equivalent inequality that is easier to work with, without altering the solution set. Mastering the art of simplification is a key skill in mathematics, applicable to a wide range of problems beyond just inequalities.
2. Isolate the Variable Term
Next, we need to isolate the term containing s on one side of the inequality. Add 3s to both sides: 12s-20 + 3s ≤ 25-3s + 3s, which simplifies to 15s-20 ≤ 25. This step is crucial for isolating the variable and moving towards a solution. By adding 3s to both sides, we effectively eliminate the s term from the right side, concentrating all terms involving s on the left. This process is guided by the principle of maintaining balance in the inequality; whatever operation is performed on one side must also be performed on the other to preserve the relationship. Isolating the variable term is a cornerstone technique in solving both equations and inequalities. It allows us to progressively narrow down the possible values of the variable until we arrive at the solution set. In the context of inequalities, this step is particularly important as it sets the stage for the final step of isolating the variable itself.
3. Isolate the Variable
To further isolate s, add 20 to both sides: 15s-20 + 20 ≤ 25 + 20, which simplifies to 15s ≤ 45. Now, divide both sides by 15: 15s / 15 ≤ 45 / 15, which gives us s ≤ 3. This is the final step in isolating the variable, revealing the solution to the inequality. By adding 20 to both sides, we eliminate the constant term on the left, bringing us closer to isolating s. Then, dividing both sides by 15 completes the process, leaving s by itself on the left side. The division operation is crucial, but it's important to remember that if we were dividing by a negative number, we would need to reverse the direction of the inequality. However, in this case, we are dividing by a positive number, so the inequality remains the same. The result, s ≤ 3, indicates that any value of s that is less than or equal to 3 will satisfy the original inequality. This solution can be represented graphically on a number line or expressed in interval notation, providing a complete understanding of the possible values of s.
Final Answer
The possible values for s in the inequality 12s-20 ≤ 50-3s-25 are s ≤ 3. Therefore, the correct answer is B) s ≤ 3. This solution represents a range of values, specifically all real numbers less than or equal to 3. Understanding that inequalities often have a range of solutions, rather than a single value, is crucial for their correct interpretation. The solution s ≤ 3 can be visualized on a number line, where a closed circle at 3 indicates that 3 is included in the solution set, and the line extends to the left, representing all values less than 3. In interval notation, the solution can be written as (-∞, 3], where the parenthesis indicates that negative infinity is not included and the square bracket indicates that 3 is included. The ability to express solutions in different forms enhances comprehension and facilitates communication in mathematical contexts. This final answer not only provides the solution to the specific problem but also reinforces the broader concept of inequality solutions and their various representations.
Common Mistakes to Avoid
When solving inequalities, it's easy to make mistakes. One common error is forgetting to reverse the inequality sign when multiplying or dividing by a negative number. For instance, if you have -2s > 4, dividing both sides by -2 should result in s < -2, not s > -2. Another mistake is incorrectly combining like terms or applying the distributive property. Always double-check your work to ensure accuracy. A thorough understanding of the properties of inequalities and careful attention to detail are key to avoiding these pitfalls. Forgetting to reverse the inequality sign when multiplying or dividing by a negative number is a critical error that can lead to an incorrect solution set. This mistake stems from a misunderstanding of how negative numbers affect the order of values. Similarly, errors in combining like terms or applying the distributive property can arise from carelessness or a lack of familiarity with algebraic techniques. These mistakes can often be avoided by practicing algebraic manipulations and developing a systematic approach to problem-solving. Another common error is failing to simplify the inequality before attempting to isolate the variable. This can lead to unnecessary complexity and increase the likelihood of mistakes. By simplifying the inequality first, you make the problem more manageable and reduce the risk of errors. In addition to these specific mistakes, a general lack of understanding of the properties of inequalities can also lead to errors. A solid grasp of these properties is essential for solving inequalities correctly and confidently.
Real-World Applications of Inequalities
Inequalities aren't just theoretical concepts; they have numerous real-world applications. For example, they can be used in budgeting to determine how much money you can spend while staying within your budget. They're also used in engineering to set safety limits and in business to model profit margins. Understanding inequalities can help you make informed decisions in everyday life. The applications of inequalities extend far beyond the classroom, touching upon various aspects of our daily lives and professional fields. In finance, inequalities are used to model investment strategies, assess risk, and determine loan eligibility. For instance, a lender might use inequalities to set criteria for loan approval, ensuring that the borrower's debt-to-income ratio falls within an acceptable range. In engineering, inequalities are crucial for designing structures and systems that can withstand specific loads and stresses. For example, engineers might use inequalities to determine the maximum weight a bridge can support or the maximum pressure a pipe can withstand. In business, inequalities are used to analyze costs, revenues, and profits, helping managers make informed decisions about pricing, production, and resource allocation. For example, a company might use inequalities to determine the minimum number of units it needs to sell to break even. Beyond these specific examples, inequalities are also used in optimization problems, where the goal is to find the best solution within a set of constraints. These constraints are often expressed as inequalities, reflecting real-world limitations such as budget constraints, resource limitations, and regulatory requirements. The ability to understand and apply inequalities is therefore a valuable skill in a wide range of contexts.
Conclusion
In conclusion, solving inequalities is a fundamental skill in mathematics with wide-ranging applications. By following the steps outlined in this article, you can confidently find the possible values for s in the inequality 12s-20 ≤ 50-3s-25. Remember to simplify, isolate the variable term, and then isolate the variable itself. Also, be mindful of common mistakes, such as forgetting to reverse the inequality sign when multiplying or dividing by a negative number. Mastering inequalities is not just about finding solutions; it's about developing a deeper understanding of mathematical relationships and their real-world implications. The journey through the process of solving 12s-20 ≤ 50-3s-25 has not only provided a specific solution but also illuminated the broader concepts and techniques involved in solving inequalities. The ability to simplify complex expressions, isolate variables, and interpret the results is a valuable skill that extends beyond the realm of mathematics. It fosters critical thinking, problem-solving, and analytical abilities that are applicable in various fields and everyday situations. By understanding inequalities, we gain a powerful tool for making informed decisions, modeling real-world scenarios, and navigating the complexities of quantitative analysis. As we conclude this discussion, it is essential to recognize that the mastery of inequalities is an ongoing process. Continuous practice, exploration of different types of inequalities, and application of these concepts to real-world problems are key to solidifying your understanding and enhancing your skills. The world of mathematics is vast and interconnected, and inequalities serve as a crucial thread in this intricate tapestry.
