Solving R/2 + 15 > 45 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. They extend the concept of equations by allowing us to express a range of possible solutions rather than a single, fixed value. Solving inequalities involves isolating the variable, similar to solving equations, but with a few key distinctions. One such inequality we'll explore in detail is: r/2 + 15 > 45. This article will provide a comprehensive guide on how to solve this specific inequality, as well as offer a broader understanding of the principles and techniques involved in solving linear inequalities in general. By the end of this guide, you'll be well-equipped to tackle a variety of inequality problems with confidence. The principles behind solving inequalities are fundamental in various fields, from economics and physics to computer science and engineering, making it a vital skill for students and professionals alike. This exploration will not only focus on the mechanical steps but also aim to provide a conceptual understanding, ensuring that you grasp the 'why' behind each step. As we progress, we will highlight common pitfalls and offer strategies to avoid them, thus solidifying your understanding and preventing errors. Furthermore, we will touch upon the graphical representation of inequality solutions, which provides a visual interpretation that can enhance your grasp of the concept. In essence, this guide aims to transform you from a mere solver of inequalities to a true understanding of their nature and application. So, let's embark on this journey of mathematical discovery and unlock the secrets of solving linear inequalities.

Understanding the Basics of Inequalities

Before diving into the specifics of solving r/2 + 15 > 45, let's establish a solid foundation by understanding the basic principles of inequalities. Inequalities, unlike equations, deal with relationships where two values are not necessarily equal. Instead, they express relationships such as 'greater than,' 'less than,' 'greater than or equal to,' or 'less than or equal to.' These relationships are represented by the symbols >, <, ≥, and ≤, respectively. Understanding these symbols and their meanings is paramount to correctly interpreting and solving inequalities. The key difference between solving equations and inequalities lies in how operations affect the solution set. While adding or subtracting the same value from both sides of an inequality works similarly to equations, multiplying or dividing by a negative number requires flipping the inequality sign. This is a crucial rule to remember, as it directly impacts the solution. Imagine an inequality as a balance scale; performing an operation on one side requires a corresponding operation on the other to maintain the balance, but multiplying or dividing by a negative number essentially flips the scale, hence the sign change. Linear inequalities, the focus of this article, involve variables raised to the power of 1. They can be represented graphically on a number line, where the solution set is often a range of values rather than a single point. This graphical representation provides a visual understanding of the solution and can be a valuable tool in verifying your answer. Furthermore, understanding the properties of inequalities, such as the transitive property (if a > b and b > c, then a > c), is essential for more complex problem-solving. With a firm grasp of these basics, we can confidently move on to solving our specific inequality and more challenging problems in the future.

Properties of Inequalities

To effectively solve inequalities, it's essential to understand their fundamental properties. These properties dictate how we can manipulate inequalities while preserving their validity. One of the most important properties is the addition and subtraction property, which states that adding or subtracting the same number from both sides of an inequality does not change the direction of the inequality. This is analogous to how we manipulate equations, making it a familiar concept. However, the multiplication and division property introduces a critical difference. When we multiply or divide both sides of an inequality by a positive number, the direction of the inequality remains the same. But, and this is crucial, when we multiply or divide by a negative number, we must flip the inequality sign. This flipping action is often a source of errors, so it's vital to pay close attention to the sign of the number you're multiplying or dividing by. Another important property is the transitive property, which states that if a > b and b > c, then a > c. This property allows us to chain inequalities together, simplifying complex relationships. For example, if we know x > y and y > z, we can directly conclude that x > z. Understanding these properties is not just about memorizing rules; it's about developing an intuitive sense of how inequalities behave. Think of an inequality as a balance scale, where multiplying or dividing by a negative number is like flipping the scale upside down. This visual analogy can help you remember the sign-flipping rule. Furthermore, these properties are not just theoretical; they have practical applications in various fields, from optimizing resource allocation in economics to determining the stability of systems in engineering. By mastering these properties, you'll be well-equipped to solve a wide range of inequality problems and apply them in real-world scenarios.

Step-by-Step Solution of r/2 + 15 > 45

Now, let's tackle the inequality r/2 + 15 > 45 step-by-step, applying the principles we've discussed. Our goal is to isolate the variable 'r' on one side of the inequality. The first step in solving this inequality is to isolate the term containing 'r'. This can be achieved by subtracting 15 from both sides of the inequality. This operation utilizes the addition/subtraction property of inequalities, which allows us to add or subtract the same value from both sides without changing the inequality's direction. Subtracting 15 from both sides gives us: r/2 + 15 - 15 > 45 - 15, which simplifies to r/2 > 30. This step effectively moves the constant term to the right side, bringing us closer to isolating 'r'. Remember, the key is to perform the same operation on both sides to maintain the balance of the inequality. The next step involves dealing with the fraction r/2. To isolate 'r', we need to eliminate the division by 2. This can be done by multiplying both sides of the inequality by 2. This is an application of the multiplication/division property of inequalities. Since we are multiplying by a positive number (2), we do not need to flip the inequality sign. Multiplying both sides by 2 gives us: 2 * (r/2) > 2 * 30, which simplifies to r > 60. This step successfully isolates 'r', giving us the solution to the inequality. The solution r > 60 means that any value of 'r' greater than 60 will satisfy the original inequality. It's important to note that 60 itself is not a solution, as the inequality is strictly 'greater than'. We can represent this solution graphically on a number line, where an open circle at 60 indicates that it's not included, and an arrow extending to the right indicates all values greater than 60 are solutions. By following these steps carefully and understanding the underlying principles, you can confidently solve similar inequalities. Always remember to check your solution by substituting a value greater than 60 into the original inequality to ensure it holds true.

Step 1: Isolate the Term with 'r'

The initial step in solving the inequality r/2 + 15 > 45 involves isolating the term containing the variable 'r'. This is a fundamental strategy in solving any algebraic equation or inequality. To achieve this, we need to eliminate the constant term, which in this case is +15, from the left side of the inequality. We accomplish this by applying the subtraction property of inequalities. This property states that subtracting the same value from both sides of an inequality preserves the inequality's direction. Therefore, we subtract 15 from both sides of the inequality. This can be written as: r/2 + 15 - 15 > 45 - 15. On the left side, the +15 and -15 cancel each other out, leaving us with just r/2. On the right side, 45 - 15 equals 30. So, after this step, the inequality simplifies to r/2 > 30. This transformation is a crucial step because it brings us closer to isolating 'r'. It's like peeling away layers to reveal the variable we're trying to solve for. This process of isolating the variable is a cornerstone of algebra, and mastering it is essential for solving more complex problems. This step also highlights the importance of performing the same operation on both sides of the inequality. This ensures that the inequality remains balanced, and the solution set remains unchanged. Think of it as maintaining equilibrium on a balance scale; whatever you do to one side, you must do to the other to keep it level. This principle applies throughout the process of solving inequalities and equations. By carefully executing this first step, we've laid a solid foundation for the subsequent steps, bringing us closer to the final solution.

Step 2: Eliminate the Fraction

Having isolated the term r/2, the next crucial step in solving the inequality r/2 > 30 is to eliminate the fraction. This means getting rid of the division by 2 that is currently affecting 'r'. To achieve this, we employ the multiplication property of inequalities. This property dictates that multiplying both sides of an inequality by the same positive number preserves the direction of the inequality. In our case, we need to multiply both sides by 2 to cancel out the division by 2 on the left side. This can be expressed as: 2 * (r/2) > 2 * 30. On the left side, multiplying r/2 by 2 results in 'r', as the 2 in the numerator and the 2 in the denominator cancel each other out. This is a fundamental arithmetic operation that simplifies the expression significantly. On the right side, 2 multiplied by 30 equals 60. Therefore, after performing this step, the inequality becomes r > 60. This is a significant milestone in solving the inequality, as we have now successfully isolated the variable 'r'. The inequality r > 60 tells us that the solution set consists of all values of 'r' that are greater than 60. It's important to note that 60 itself is not included in the solution set, as the inequality is strictly 'greater than'. This distinction is crucial when interpreting the solution and representing it graphically. This step also underscores the importance of understanding how operations interact with fractions in algebraic expressions. By mastering these fundamental skills, you'll be well-equipped to tackle more complex problems involving fractions and inequalities. With 'r' now isolated, we have effectively solved the inequality, but it's always a good practice to verify the solution and understand its implications.

Solution and Interpretation

After performing the steps of isolating the term with 'r' and eliminating the fraction, we arrive at the solution r > 60. This solution represents a set of values, not just a single value, that satisfy the original inequality r/2 + 15 > 45. Interpreting this solution is crucial to fully understanding its meaning and implications. The inequality r > 60 states that 'r' can be any number greater than 60, but not including 60 itself. This is a key distinction; the '>' symbol means 'strictly greater than'. If the inequality were r ≥ 60, then 60 would be included in the solution set. To visualize this solution, we can represent it on a number line. Draw a number line and mark the point 60. Since 60 is not included, we use an open circle at 60. Then, we draw an arrow extending to the right from 60, indicating that all values to the right of 60 are part of the solution set. This visual representation provides a clear and intuitive understanding of the solution. It's also important to verify the solution to ensure its correctness. We can do this by choosing a value greater than 60 and substituting it back into the original inequality. For example, let's choose r = 70. Substituting this into the original inequality, we get: 70/2 + 15 > 45, which simplifies to 35 + 15 > 45, and further to 50 > 45. This is a true statement, confirming that 70 is indeed a solution. Conversely, if we choose a value less than or equal to 60, such as r = 60, the inequality does not hold true: 60/2 + 15 > 45 simplifies to 30 + 15 > 45, which becomes 45 > 45, a false statement. This further validates our solution. Understanding the solution and its interpretation is not just about finding the answer; it's about grasping the underlying concept and being able to apply it in different contexts. This skill is essential for problem-solving in various fields, from mathematics and physics to economics and engineering.

Graphical Representation of the Solution

Visualizing the solution of an inequality can significantly enhance understanding. The graphical representation of the solution r > 60 is a number line with a specific marking to indicate the solution set. To construct this graphical representation, start by drawing a number line. This line represents all real numbers, extending infinitely in both directions. Mark the point 60 on the number line. This point is the critical value that separates the solutions from the non-solutions. Now, we need to indicate whether 60 itself is included in the solution set. Since our inequality is r > 60, which means 'r is strictly greater than 60', we use an open circle at the point 60. An open circle signifies that the value is not included in the solution. If the inequality were r ≥ 60 (r is greater than or equal to 60), we would use a closed circle or a filled-in circle to indicate that 60 is part of the solution. Next, we need to represent all the values that are greater than 60. This is done by drawing an arrow extending to the right from the open circle at 60. This arrow indicates that all numbers to the right of 60, up to positive infinity, are solutions to the inequality. The graphical representation provides a clear and intuitive picture of the solution set. It allows you to quickly see the range of values that satisfy the inequality. This visualization is particularly helpful when dealing with more complex inequalities or systems of inequalities. For example, if we had another inequality, such as r < 70, we could represent it on the same number line with an open circle at 70 and an arrow extending to the left. The overlap between the two solution sets would then represent the solution to the system of inequalities. Graphical representation is not just a visual aid; it's a powerful tool for problem-solving. It can help you identify errors, understand the nature of the solution, and communicate your results effectively. By mastering this technique, you'll gain a deeper understanding of inequalities and their applications.

Common Mistakes to Avoid

When solving inequalities, it's easy to make mistakes if you're not careful. Identifying common pitfalls can help you avoid errors and ensure accurate solutions. One of the most frequent mistakes is forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number. This is a crucial rule that must be followed to maintain the validity of the inequality. Remember, multiplying or dividing by a negative number essentially reverses the order of the number line, hence the sign flip. Another common mistake is treating inequalities like equations and assuming that you can always perform the same operations without any changes. While many operations are similar, the sign-flipping rule is a significant difference. A third mistake is misinterpreting the inequality symbols. It's essential to understand the difference between >, <, ≥, and ≤. The symbols > and < represent 'strictly greater than' and 'strictly less than', respectively, meaning the endpoint is not included in the solution. The symbols ≥ and ≤ represent 'greater than or equal to' and 'less than or equal to', respectively, meaning the endpoint is included. Misinterpreting these symbols can lead to incorrect solutions and graphical representations. Another pitfall is not verifying the solution. Always check your answer by substituting a value from your solution set back into the original inequality. This will help you catch any errors you may have made along the way. Additionally, when dealing with compound inequalities (e.g., a < x < b), it's important to keep the entire inequality intact and perform operations on all parts. Breaking it into separate inequalities can lead to confusion and errors. By being aware of these common mistakes and taking steps to avoid them, you can significantly improve your accuracy in solving inequalities. Practice and attention to detail are key to mastering this skill.

Conclusion

In conclusion, solving the inequality r/2 + 15 > 45 involves a series of steps based on the fundamental properties of inequalities. We began by isolating the term containing the variable 'r' by subtracting 15 from both sides, resulting in r/2 > 30. Then, we eliminated the fraction by multiplying both sides by 2, leading to the solution r > 60. This solution signifies that any value of 'r' greater than 60 will satisfy the original inequality. We emphasized the importance of understanding the properties of inequalities, particularly the rule of flipping the inequality sign when multiplying or dividing by a negative number. We also discussed the graphical representation of the solution on a number line, where an open circle at 60 and an arrow extending to the right visually represent the solution set. Furthermore, we highlighted common mistakes to avoid, such as forgetting to flip the sign or misinterpreting the inequality symbols. By understanding these principles and avoiding these pitfalls, you can confidently solve a wide range of linear inequalities. The ability to solve inequalities is a valuable skill in mathematics and has applications in various fields, from science and engineering to economics and finance. It allows us to model and analyze situations where values are not necessarily equal but fall within a certain range. As you continue your mathematical journey, remember that practice and attention to detail are key to mastering any skill, including solving inequalities. By consistently applying the steps and principles discussed in this article, you'll develop a strong foundation in this area and be well-prepared to tackle more complex problems in the future. The journey of mathematical learning is one of continuous growth and discovery, and mastering inequalities is a significant step along that path. So, embrace the challenge, practice diligently, and unlock the power of inequalities!