Solving Inequalities Step-by-Step Guide For (r/2) + 15 > 45

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In this comprehensive guide, we will delve into the process of solving the inequality ${ \frac{r}{2}+15 > 45 }$. Inequalities, a fundamental concept in mathematics, play a crucial role in various fields, including algebra, calculus, and optimization. Mastering the techniques for solving inequalities is essential for students, researchers, and professionals alike. This article aims to provide a clear, step-by-step approach to understanding and solving this particular inequality, ensuring that you grasp the underlying principles and can apply them to other similar problems.

Before we dive into the specifics of solving the inequality ${ \frac{r}{2}+15 > 45 }$, it's crucial to understand what inequalities are and how they differ from equations. An inequality is a mathematical statement that compares two expressions using inequality symbols such as greater than (>), less than (<), greater than or equal to (\≥), and less than or equal to (\≤). Unlike equations, which assert the equality of two expressions, inequalities indicate a range of possible values that satisfy the given condition. Understanding these fundamental differences is crucial for correctly manipulating and solving inequalities. Inequalities, in their essence, are comparisons that define a range of solutions rather than a single solution. The greater than symbol, represented as '>,' indicates that the left-hand side of the inequality is larger than the right-hand side. Similarly, the less than symbol, '<,' signifies that the left-hand side is smaller than the right-hand side. The symbols '≥' and '≤' introduce an added layer of precision, indicating 'greater than or equal to' and 'less than or equal to,' respectively. This subtle but significant distinction expands the solution set to include the endpoint value, which is pivotal in numerous mathematical contexts. When solving inequalities, it's paramount to understand that the operations performed on both sides must preserve the truth of the inequality. Certain operations, like multiplying or dividing by a negative number, necessitate a reversal of the inequality sign to maintain the correctness of the statement. For example, if we have the inequality -x > 5, multiplying both sides by -1 requires us to flip the inequality sign, resulting in x < -5. This rule is critical and often a point of error for many students. Another important aspect to remember when dealing with inequalities is the representation of solutions. The solution to an inequality is often not a single number, but a range of numbers. This range can be expressed in several ways: using inequality notation (e.g., x > 3), interval notation (e.g., (3, ∞)), or graphically on a number line. Understanding how to convert between these representations is a crucial skill in mathematics. Inequalities are not confined to simple linear forms; they can involve polynomials, rational expressions, and even absolute values. Each type of inequality requires a specific approach. For instance, solving quadratic inequalities often involves finding the roots of the corresponding quadratic equation and testing intervals to determine where the inequality holds true. Rational inequalities, on the other hand, require identifying critical points, which include the roots of the numerator and denominator, and then using these points to divide the number line into intervals for testing. Absolute value inequalities introduce another layer of complexity, as the absolute value of a quantity can be interpreted as its distance from zero. Therefore, an inequality like |x| < 3 implies that x is within 3 units of zero, leading to the solution -3 < x < 3. The principles of inequalities extend far beyond textbook exercises, finding practical applications in various real-world scenarios. In economics, inequalities are used to model constraints and optimize resources. In engineering, they are essential for setting tolerances and ensuring safety margins. In computer science, inequalities are used in algorithms for sorting, searching, and optimization. Moreover, understanding inequalities helps in making informed decisions in everyday life, such as budgeting, time management, and even assessing risks and rewards. The ability to interpret and solve inequalities is a valuable skill that empowers individuals to approach problems analytically and make sound judgments.

The initial step in solving the inequality **$ \frac{r}{2}+15 > 45 }** is to isolate the term containing the variable **_r_**. This involves eliminating any constants that are added to or subtracted from the term. In our case, we have **+15** on the left side of the inequality. To remove this, we perform the inverse operation, which is subtraction. We subtract 15 from both sides of the inequality to maintain the balance and ensure that the inequality remains valid. The core principle behind solving inequalities, much like solving equations, is the concept of maintaining balance. An inequality essentially represents a relationship between two expressions, and any operation performed must preserve this relationship. This means that whatever action you take on one side of the inequality, you must also apply to the other side. In the context of isolating a variable, the goal is to methodically remove any constants or coefficients that are obstructing the variable's clarity. The first step typically involves addressing addition or subtraction. If a constant is added to the term containing the variable, you subtract that constant from both sides. Conversely, if a constant is subtracted, you add it to both sides. This process ensures that the variable term is progressively isolated, bringing us closer to the solution. In our specific case, we begin with the inequality \${ \\frac{r}{2} + 15 > 45 \}. The presence of '+15' on the left side calls for its removal to isolate the term ${ \frac{r}{2} }$. To achieve this, we apply the principle of inverse operations, subtracting 15 from both sides of the inequality. This action maintains the balance of the inequality and allows us to simplify the expression. Subtracting 15 from both sides is not an arbitrary step; it is a deliberate action rooted in the properties of inequalities. By performing the same operation on both sides, we ensure that the relationship '>' remains valid. This is a critical aspect of inequality manipulation, as violating this principle can lead to incorrect solutions. After subtracting 15 from both sides, we obtain a simplified inequality ${ \frac{r{2} > 30 }$. This new inequality is mathematically equivalent to the original but is now in a form that brings us one step closer to isolating the variable r. The term containing r is now more isolated, with only the division by 2 remaining to be addressed. The subtraction of 15 is not just a mechanical step; it is a logical progression that simplifies the inequality while preserving its inherent meaning. It showcases the power of inverse operations in unraveling mathematical expressions and highlights the importance of maintaining balance in inequality manipulations. Moreover, this step underscores the iterative nature of problem-solving in mathematics, where complex problems are broken down into smaller, more manageable steps. By systematically addressing each component, we pave the way for a clear and accurate solution. This methodical approach is not only effective in solving inequalities but also serves as a valuable strategy for tackling a wide range of mathematical challenges. As we progress through the subsequent steps, we will continue to apply similar principles, each action building upon the previous one, to ultimately arrive at the solution for r.

Having successfully isolated the term **${ \frac{r}{2} }**, the next step is to eliminate the denominator, which is 2 in this case. To achieve this, we multiply both sides of the inequality by 2. This operation is based on the principle that multiplying both sides of an inequality by a positive number preserves the inequality's direction. Multiplying by 2 effectively cancels out the division by 2, leaving us with **_r_** isolated on the left side. The principle of maintaining balance in inequalities is paramount when performing operations on both sides. In the previous step, we addressed addition and subtraction by using their inverse operations. Similarly, when confronted with multiplication or division, we employ their respective inverses to isolate the variable. In our current scenario, the variable **_r_** is being divided by 2. To counteract this division, we must multiply both sides of the inequality by 2. This action is not arbitrary; it is a deliberate application of mathematical principles aimed at simplifying the expression and bringing us closer to the solution. When multiplying both sides of an inequality, it is crucial to consider the sign of the multiplier. Multiplying by a positive number, as we are doing here with 2, preserves the direction of the inequality. However, multiplying by a negative number necessitates a reversal of the inequality sign to maintain the truth of the statement. This is a critical distinction that must be carefully observed to avoid errors in the solution. In our specific case, since we are multiplying by the positive number 2, we do not need to reverse the inequality sign. We start with the inequality \${ \\frac{r}{2} > 30 \}. By multiplying both sides by 2, we are essentially undoing the division by 2, which will isolate r on the left side. This step is a fundamental algebraic manipulation that simplifies the inequality and allows us to determine the range of values for r that satisfy the original condition. The multiplication by 2 is not just a mechanical operation; it is a logical step that removes a barrier to understanding the value of r. It transforms the inequality into a more transparent form, where the relationship between r and the numerical value becomes clearer. This process highlights the power of algebraic manipulations in simplifying complex expressions and revealing underlying relationships. Moreover, this step underscores the sequential nature of problem-solving in mathematics, where each action builds upon the previous one. By methodically addressing each component of the inequality, we pave the way for a clear and accurate solution. The multiplication step is a testament to the elegance and efficiency of mathematical methods in unraveling complex problems. As we move forward, we will see how this step has set the stage for the final determination of the solution set for r. The process of multiplying both sides by 2 is not just about getting to the answer; it's about understanding the underlying principles and the logical flow of mathematical reasoning.

After performing the multiplication, we arrive at the simplified inequality $ r > 60 }**. This inequality states that **_r_** is greater than 60. This is the solution to the original inequality. It means that any value of **_r_** that is greater than 60 will satisfy the inequality **\${ \\frac{r}{2}+15 > 45 \}**. The culmination of our efforts in solving the inequality ${ \frac{r}{2} + 15 > 45 }$ leads us to the concise and definitive solution ${ r > 60 }**. This inequality is not merely an answer; it is a statement that encapsulates the entire range of values for **_r_** that satisfy the initial condition. It signifies that any number greater than 60, when substituted for **_r_** in the original inequality, will make the statement true. The beauty of mathematical solutions lies in their precision and generality. The inequality **\${ r > 60 \} is precise because it clearly defines the boundary of the solution set: 60. It is general because it applies to an infinite number of values; any number greater than 60 is a valid solution. This contrasts sharply with equations, which typically have a finite number of solutions, often just one. The solution ${ r > 60 }** can be visualized in several ways, each offering a different perspective on its meaning. One common method is the number line, where we represent the solution as an open interval extending from 60 to infinity. The open circle at 60 indicates that 60 itself is not included in the solution set, as the inequality is strictly 'greater than' and not 'greater than or equal to.' Another way to interpret the solution is in the context of the original problem. If the inequality represented a real-world scenario, such as a minimum requirement or a threshold to be surpassed, then the solution **\${ r > 60 \} would provide a clear guideline. For example, if r represented the number of units that needed to be sold to achieve a certain profit, then the solution would tell us that more than 60 units must be sold to reach the target. The process of arriving at the solution ${ r > 60 }** is a testament to the power of algebraic manipulation and the importance of following logical steps. Each step, from isolating the variable term to multiplying by the appropriate factor, was a deliberate action designed to simplify the inequality while preserving its fundamental meaning. The solution **\${ r > 60 \} is not just the end of the problem; it is a starting point for further exploration. It can be used to test specific values, to make predictions, and to gain deeper insights into the relationship between variables. Moreover, the process of solving this inequality provides a template for tackling other inequalities, demonstrating the generalizability of mathematical methods. In conclusion, the solution ${ r > 60 }$ is a concise, precise, and general statement that answers the question posed by the inequality **${ \frac{r{2} + 15 > 45 }$. It is a testament to the power of mathematical reasoning and the elegance of algebraic solutions.

In this detailed guide, we have successfully solved the inequality $ \frac{r}{2}+15 > 45 }**, arriving at the solution **\${ r > 60 \}**. This process involved isolating the variable r through a series of algebraic manipulations, including subtraction and multiplication. Understanding these steps is crucial for solving a wide range of inequalities. The journey through solving the inequality ${ \frac{r}{2} + 15 > 45 }$ has been a comprehensive exploration of the principles and techniques involved in handling inequalities. We began by establishing a solid understanding of what inequalities are and how they differ from equations, emphasizing the importance of maintaining balance and adhering to the rules of inequality manipulation. This foundational knowledge set the stage for the subsequent steps, ensuring that the solution we arrived at was not only correct but also deeply understood. The first step in our journey involved isolating the term containing the variable r. This was achieved by strategically subtracting 15 from both sides of the inequality. This action, rooted in the principle of inverse operations, demonstrated the power of methodical simplification in unraveling mathematical expressions. By removing the constant term, we brought the variable closer to isolation, paving the way for the next phase of the solution. The second step focused on eliminating the denominator that was dividing the variable. Multiplying both sides of the inequality by 2 effectively counteracted the division, bringing us closer to a direct understanding of the value of r. This step highlighted the importance of considering the sign of the multiplier and its effect on the direction of the inequality, a crucial aspect often overlooked by those new to inequality manipulation. Finally, we arrived at the solution ${ r > 60 }$. This inequality is more than just an answer; it's a clear and concise statement that defines the range of values for r that satisfy the original condition. The solution was presented not just as a final result but as a starting point for further exploration and understanding. It can be visualized on a number line, interpreted in the context of real-world scenarios, and used to test specific values, demonstrating the versatility and applicability of mathematical solutions. Throughout this process, we have emphasized not just the mechanics of solving the inequality but also the underlying principles and logical reasoning that make the solution meaningful. We have highlighted the importance of maintaining balance, using inverse operations, and carefully considering the effects of each manipulation on the inequality. This holistic approach ensures that the skills learned are not just procedural but also deeply conceptual, allowing for greater flexibility and problem-solving ability in future mathematical endeavors. In conclusion, solving the inequality **${ \frac{r{2} + 15 > 45 }** has been a journey through the fundamentals of inequality manipulation. The solution **\${ r > 60 \} is a testament to the power of methodical problem-solving and the elegance of algebraic methods. This experience provides a solid foundation for tackling more complex inequalities and serves as a valuable tool in various mathematical and real-world applications. The skills acquired are not just about solving one specific problem but about developing a broader understanding of mathematical principles and their application to problem-solving.

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