Solving The Inequality (2x + 1)/4 > (x + 3)/3 A Step-by-Step Guide

In this article, we will delve into the process of solving the inequality (2x + 1)/4 > (x + 3)/3. Inequalities, a fundamental concept in mathematics, are mathematical statements that compare two expressions using inequality symbols such as '>', '<', '≥', or '≤'. Solving inequalities involves finding the range of values for a variable that makes the inequality true. This skill is crucial in various fields, including algebra, calculus, and real-world problem-solving. This exploration of solving linear inequalities is a key skill in algebra, offering a foundation for more complex mathematical concepts and applications. Understanding how to manipulate and solve these inequalities is not just a mathematical exercise; it's a tool that allows us to model and solve problems in various real-world scenarios, from optimizing resources to understanding constraints in engineering and economics. Linear inequalities, specifically, involve comparing linear expressions, which are expressions where the variable is raised to the power of one. These inequalities are a cornerstone of mathematical analysis and are frequently encountered in both theoretical and practical contexts. In the following sections, we will systematically break down the steps involved in solving the given inequality, providing clear explanations and justifications for each step. By mastering the techniques presented here, you will be well-equipped to tackle a wide range of inequality problems and gain a deeper understanding of their significance in mathematics and beyond. So, let's embark on this mathematical journey and unravel the solution to the inequality (2x + 1)/4 > (x + 3)/3. This article aims to provide a comprehensive understanding of the step-by-step process involved in solving the inequality, ensuring that readers can confidently tackle similar problems in the future. We will begin by eliminating the fractions, then simplify the inequality, and finally isolate the variable to find the solution set. The methodology employed here is not only applicable to this specific problem but can also be generalized to solve a broad spectrum of linear inequalities. Throughout the discussion, emphasis will be placed on the underlying principles and reasoning behind each step, fostering a deeper understanding of the mathematical concepts involved. This approach will empower readers to not only solve inequalities mechanically but also to comprehend the logic and rationale behind the solution process.

1. Eliminating Fractions

To solve the inequality (2x + 1)/4 > (x + 3)/3, the initial step involves eliminating the fractions. Fractions in inequalities can complicate the solution process, making it harder to isolate the variable. To remove these fractions, we need to find the least common multiple (LCM) of the denominators, which in this case are 4 and 3. The LCM of 4 and 3 is 12. Multiplying both sides of the inequality by the LCM will clear the fractions, transforming the inequality into a simpler form that is easier to solve. This is a fundamental technique in solving inequalities and equations, as it allows us to work with whole numbers instead of fractions, which simplifies the subsequent algebraic manipulations. Remember, when multiplying both sides of an inequality by a positive number, the direction of the inequality sign remains the same. However, if we were to multiply by a negative number, we would need to reverse the inequality sign. In this case, since we are multiplying by a positive number (12), the inequality sign will not change. This step is crucial because it sets the stage for the rest of the solution. By eliminating the fractions, we pave the way for a more straightforward simplification process. This principle of clearing fractions is a universally applicable technique in algebra and is not limited to just inequalities. It can be used to solve equations, systems of equations, and other algebraic problems involving fractions. By mastering this step, you will be able to tackle a wider range of mathematical problems with greater ease and confidence. So, let's proceed with multiplying both sides of the inequality by 12, which will effectively eliminate the fractions and allow us to move on to the next phase of solving the inequality. This process of eliminating fractions is akin to clearing obstacles from a path, making the journey to the solution much smoother and more efficient. It's a testament to the power of basic algebraic techniques in simplifying complex problems. Once the fractions are gone, we can focus on the core of the inequality, manipulating the terms to isolate the variable and find the solution set. This is where the real algebraic work begins, and we will see how the simple act of eliminating fractions sets the stage for a more streamlined solution process. The elimination of fractions is not just a mechanical step; it's a strategic move that significantly reduces the complexity of the problem, making it more accessible and manageable. It's a perfect example of how a well-chosen algebraic manipulation can make a world of difference in solving mathematical problems.

2. Simplifying the Inequality

After multiplying both sides of the inequality (2x + 1)/4 > (x + 3)/3 by 12, we obtain the equation 3(2x + 1) > 4(x + 3). This equation is now free of fractions, which makes it significantly easier to work with. The next step in solving the inequality involves simplifying both sides of the equation. Simplification is a crucial process in algebra as it reduces the complexity of expressions and equations, making them more manageable and easier to understand. In this case, simplification involves applying the distributive property to expand the expressions on both sides of the inequality. The distributive property states that a(b + c) = ab + ac, which means we multiply the term outside the parentheses by each term inside the parentheses. Applying the distributive property on the left side, we multiply 3 by both 2x and 1, resulting in 6x + 3. Similarly, on the right side, we multiply 4 by both x and 3, yielding 4x + 12. Therefore, the inequality becomes 6x + 3 > 4x + 12. This simplified form is much easier to handle than the original inequality with fractions. The process of simplification is not just about making the equation look simpler; it's about revealing the underlying structure and relationships between the terms. By expanding the expressions, we make the terms involving the variable (x) and the constant terms more visible, which is essential for the next step of isolating the variable. This simplification step is a testament to the power of algebraic manipulation in transforming complex expressions into simpler, more manageable forms. It's a fundamental skill in algebra that is used extensively in solving equations, inequalities, and various other mathematical problems. By mastering the art of simplification, you can significantly enhance your problem-solving abilities and gain a deeper understanding of mathematical concepts. The simplified inequality, 6x + 3 > 4x + 12, now presents a clearer path towards isolating the variable x. We have successfully navigated the initial hurdle of fractions and have now arrived at a form that is conducive to further algebraic manipulation. This step highlights the importance of breaking down complex problems into smaller, more manageable steps. By tackling each step methodically, we can unravel even the most challenging mathematical problems. The simplification process also demonstrates the interconnectedness of algebraic concepts. The distributive property, a fundamental principle in algebra, plays a crucial role in simplifying expressions and equations. Understanding and applying these principles effectively is key to success in mathematics.

3. Isolating the Variable

Having simplified the inequality to 6x + 3 > 4x + 12, the next crucial step is to isolate the variable 'x'. Isolating the variable means getting 'x' by itself on one side of the inequality. This is achieved by performing a series of algebraic operations that move all terms containing 'x' to one side and all constant terms to the other side. The key principle here is to maintain the balance of the inequality. Any operation performed on one side must also be performed on the other side to ensure that the inequality remains valid. To begin isolating 'x', we can subtract 4x from both sides of the inequality. This eliminates the '4x' term on the right side and moves the 'x' terms to the left side. Subtracting 4x from both sides gives us: 6x + 3 - 4x > 4x + 12 - 4x, which simplifies to 2x + 3 > 12. Now, we have all the 'x' terms on the left side, but we still have a constant term (+3) on the same side. To move this constant term to the right side, we subtract 3 from both sides of the inequality. This gives us: 2x + 3 - 3 > 12 - 3, which simplifies to 2x > 9. We are now very close to isolating 'x'. We have 2x on the left side, which means 'x' is being multiplied by 2. To undo this multiplication and get 'x' by itself, we divide both sides of the inequality by 2. Dividing both sides by 2 gives us: (2x)/2 > 9/2, which simplifies to x > 9/2. This is the final step in isolating the variable. We have successfully manipulated the inequality to get 'x' by itself on one side, with the inequality sign and a constant on the other side. The result, x > 9/2, tells us that the solution to the inequality is all values of 'x' that are greater than 9/2. Isolating the variable is a fundamental technique in solving equations and inequalities. It's a process that requires careful application of algebraic principles and a clear understanding of how operations affect the balance of the equation or inequality. By mastering this technique, you can solve a wide range of mathematical problems and gain a deeper understanding of algebraic concepts. The process of isolating the variable is not just a mechanical procedure; it's a logical and systematic approach to solving mathematical problems. Each step is carefully chosen to bring us closer to the solution, and the principles of algebra ensure that the inequality remains valid throughout the process. The result, x > 9/2, is not just a number; it's a set of values that satisfy the original inequality. This highlights the difference between solving equations and inequalities. Equations typically have a single solution, while inequalities have a range of solutions.

4. Expressing the Solution

After isolating the variable, we arrive at the solution x > 9/2. This inequality tells us that the solution set includes all values of 'x' that are greater than 9/2. However, expressing the solution in different forms can provide a more complete understanding and facilitate its use in various contexts. There are several ways to express the solution, including graphically on a number line, in interval notation, and in set notation. Each of these representations offers a unique perspective on the solution set and can be useful in different situations. First, let's consider the graphical representation on a number line. A number line is a visual tool that represents all real numbers as points on a line. To represent the solution x > 9/2 on a number line, we first locate the point 9/2 (which is 4.5) on the line. Since the inequality is 'greater than' (>) and not 'greater than or equal to' (≥), we use an open circle at 9/2 to indicate that 9/2 is not included in the solution set. Then, we draw an arrow extending to the right from the open circle, indicating that all numbers to the right of 9/2 are part of the solution. This visual representation provides a clear and intuitive understanding of the solution set. Next, let's express the solution in interval notation. Interval notation is a way of writing solution sets using intervals. An interval is a set of numbers between two endpoints, which can be finite or infinite. For the inequality x > 9/2, the solution set extends from 9/2 to positive infinity. In interval notation, we write this as (9/2, ∞). The parenthesis '(' indicates that 9/2 is not included in the interval, which is consistent with the '>' inequality. The symbol '∞' represents positive infinity, and we always use a parenthesis with infinity since infinity is not a specific number and cannot be included in the interval. Interval notation is a concise and precise way of expressing solution sets, especially when dealing with inequalities. Finally, let's express the solution in set notation. Set notation is a formal way of defining a set using mathematical symbols. For the inequality x > 9/2, we can write the solution set in set notation as {x | x > 9/2}. This is read as "the set of all x such that x is greater than 9/2." The curly braces '{}' denote a set, the vertical bar '|' is read as "such that," and the inequality x > 9/2 defines the condition for membership in the set. Set notation provides a rigorous and unambiguous way of defining solution sets and is commonly used in more advanced mathematical contexts. Each of these representations – graphical, interval, and set notation – offers a different way of expressing the solution x > 9/2. The choice of representation depends on the context and the specific needs of the problem. Understanding these different representations enhances our ability to communicate mathematical ideas effectively and to solve problems in a variety of situations. The solution to the inequality is not just a number; it's a set of numbers that satisfy a certain condition. Expressing this solution in different forms allows us to fully grasp its meaning and to use it in various applications.

5. Conclusion

In conclusion, we have successfully solved the inequality (2x + 1)/4 > (x + 3)/3 by systematically applying algebraic principles. The step-by-step process involved eliminating fractions, simplifying the inequality, isolating the variable, and expressing the solution in various forms. This journey through the solution process has not only provided us with the answer but has also reinforced our understanding of fundamental algebraic concepts and techniques. The solution, x > 9/2, represents a set of values that satisfy the inequality. We have seen how this solution can be expressed graphically on a number line, in interval notation as (9/2, ∞), and in set notation as {x | x > 9/2}. Each representation offers a unique perspective on the solution set and is valuable in different contexts. The ability to solve inequalities is a crucial skill in mathematics and has wide-ranging applications in various fields. Inequalities are used to model and solve problems involving constraints, optimization, and comparisons. They are essential tools in areas such as engineering, economics, computer science, and physics. The techniques we have employed in solving this inequality are not limited to this specific problem. They can be generalized to solve a broad spectrum of linear inequalities. The key is to approach the problem methodically, applying the principles of algebra consistently and carefully. The process of solving inequalities is not just about finding the answer; it's about developing problem-solving skills and deepening our understanding of mathematical concepts. Each step in the solution process – eliminating fractions, simplifying, isolating the variable – requires careful reasoning and application of algebraic principles. By mastering these skills, we become more confident and capable mathematicians. The solution x > 9/2 is not the end of the story; it's a starting point for further exploration and application. We can use this solution to solve related problems, to make predictions, and to gain insights into real-world phenomena. Mathematics is not just a collection of rules and formulas; it's a powerful tool for understanding and shaping the world around us. By embracing the challenges of mathematics and by mastering fundamental skills like solving inequalities, we empower ourselves to tackle complex problems and to make meaningful contributions to society. The journey of solving the inequality (2x + 1)/4 > (x + 3)/3 has been a journey of mathematical discovery and empowerment. We have not only found the solution but have also reinforced our understanding of the underlying principles and techniques. This understanding will serve us well in future mathematical endeavors and in various aspects of our lives. The power of mathematics lies not just in its ability to solve problems but in its ability to transform the way we think and approach the world. By engaging with mathematics in a thoughtful and systematic way, we develop critical thinking skills, problem-solving abilities, and a deep appreciation for the beauty and elegance of mathematical concepts.