Solving \(\frac{r}{2}-4 < 19\) A Step-by-Step Guide

Introduction: Understanding Inequalities

In the realm of mathematics, inequalities play a crucial role in expressing relationships where values are not necessarily equal. Unlike equations that assert equality, inequalities describe situations where one value is greater than, less than, greater than or equal to, or less than or equal to another value. Mastering the art of solving inequalities is fundamental for various mathematical applications, ranging from algebra to calculus and beyond. This comprehensive guide will delve into the intricacies of solving a specific inequality, ${\frac{r}{2}-4 < 19}$, providing a step-by-step approach that demystifies the process and empowers you to tackle similar problems with confidence.

The foundation of inequality manipulation lies in the understanding that certain operations can be performed on both sides of the inequality without altering its fundamental truth. These operations include adding or subtracting the same value from both sides, as well as multiplying or dividing both sides by a positive value. However, a crucial caveat exists when multiplying or dividing by a negative value, as this necessitates flipping the direction of the inequality sign to maintain the integrity of the relationship. To solve the inequality, our primary objective is to isolate the variable r on one side of the inequality. This involves strategically applying the aforementioned operations to systematically eliminate constants and coefficients surrounding the variable. Each step taken must adhere to the fundamental principles of inequality manipulation, ensuring that the solution obtained accurately represents the range of values that satisfy the original inequality. Before embarking on the step-by-step solution, it's essential to grasp the significance of the solution set. The solution to an inequality is not a single value, as in the case of an equation, but rather a range of values that satisfy the given condition. This range can be represented graphically on a number line, providing a visual depiction of the solution set. For instance, if the solution is r > 5, then all values greater than 5 are included in the solution set, represented by an open interval extending infinitely to the right on the number line. Understanding the solution set is crucial for interpreting the meaning of the inequality and its implications in various contexts.

Step 1: Isolating the Variable Term

The first crucial step in solving the inequality ${\frac{r}{2}-4 < 19}$ is to isolate the term containing the variable r. In this case, the variable term is ${\frac{r}{2}}$. To achieve isolation, we need to eliminate the constant term, which is -4, from the left side of the inequality. The fundamental principle of inequality manipulation dictates that we can add the same value to both sides of the inequality without altering its direction or truth. Therefore, to eliminate the -4, we add 4 to both sides of the inequality. This operation effectively cancels out the -4 on the left side, leaving us with the variable term alone. The mathematical representation of this step is as follows:

${ \frac{r}{2} - 4 + 4 < 19 + 4 }$

Simplifying the equation, we obtain:

${ \frac{r}{2} < 23 }$

This step represents a significant milestone in solving the inequality. By isolating the variable term, we have brought ourselves closer to determining the range of values for r that satisfy the original inequality. The inequality ${\frac{r}{2} < 23}$ now presents a clearer picture of the relationship between r and the constant value 23. However, the variable r is still encumbered by the coefficient ${\frac{1}{2}}$. The next step will address this coefficient, further isolating r and ultimately revealing the solution set.

The importance of this initial step cannot be overstated. It lays the groundwork for the subsequent steps by simplifying the inequality and focusing our attention on the variable term. The addition property of inequality, which allows us to add the same value to both sides without changing the inequality's direction, is a cornerstone of inequality manipulation. Mastering this principle is essential for solving a wide range of inequalities. Furthermore, it's crucial to maintain a clear and organized approach throughout the solving process. Each step should be clearly documented, and the reasoning behind each operation should be understood. This not only ensures accuracy but also facilitates the learning process and allows for easier error detection if needed. In summary, the first step of isolating the variable term by adding 4 to both sides of the inequality is a fundamental step that sets the stage for the subsequent steps and ultimately leads to the solution of the inequality.

Step 2: Solving for r

Having successfully isolated the variable term ${\frac{r}{2}}$ in the previous step, the next logical step is to solve for r itself. The inequality we currently have is ${\frac{r}{2} < 23}$. To isolate r, we need to eliminate the denominator 2 from the left side of the inequality. The principle that governs this operation is the multiplication property of inequality. This property states that we can multiply both sides of an inequality by the same positive value without altering the direction of the inequality sign. In this case, we need to multiply both sides by 2. The mathematical representation of this step is as follows:

${ 2 \times \frac{r}{2} < 23 \times 2 }$

Performing the multiplication on both sides, we get:

${ r < 46 }$

This is the solution to the inequality. It states that r is less than 46. This means that any value of r that is strictly less than 46 will satisfy the original inequality ${\frac{r}{2}-4 < 19}$. The solution set is an infinite range of values, extending from negative infinity up to, but not including, 46.

To fully understand the solution, it's helpful to visualize it on a number line. Imagine a number line extending from negative infinity to positive infinity. The solution r < 46 is represented by an open interval extending from negative infinity up to 46. The open circle at 46 indicates that 46 itself is not included in the solution set. Any point on the number line to the left of 46 represents a value of r that satisfies the inequality. This visual representation provides a clear and intuitive understanding of the solution set. It's crucial to note that the multiplication property of inequality has a critical caveat. If we were to multiply both sides of the inequality by a negative number, we would need to flip the direction of the inequality sign. This is because multiplying by a negative number reverses the order of the numbers on the number line. For example, if we had -r < 5, multiplying both sides by -1 would result in r > -5, with the inequality sign flipped. In this particular case, we multiplied by a positive number (2), so the inequality sign remained unchanged. This step of solving for r by multiplying both sides of the inequality by 2 is a pivotal moment in the solution process. It reveals the explicit relationship between r and the constant value 46, allowing us to define the solution set precisely. The solution r < 46 provides a concise and unambiguous answer to the original inequality problem. The ability to confidently apply the multiplication property of inequality, while being mindful of the sign-flipping rule, is an essential skill in solving inequalities.

Step 3: Verifying the Solution

After obtaining the solution r < 46, it's crucial to verify its correctness. This step ensures that we haven't made any errors in our calculations and that the solution accurately represents the values that satisfy the original inequality ${\frac{r}{2}-4 < 19}$. Verification involves substituting values from the solution set back into the original inequality and checking if the inequality holds true. To perform the verification, we need to select a value that is less than 46. Let's choose a value that is significantly less than 46, such as r = 30. This choice provides a good test of the solution, as it's far enough from the boundary value of 46. Substituting r = 30 into the original inequality, we get:

${ \frac{30}{2} - 4 < 19 }$

Simplifying the left side of the inequality:

${ 15 - 4 < 19 }$

${ 11 < 19 }$

The inequality 11 < 19 is indeed true. This confirms that r = 30, a value within our solution set, satisfies the original inequality. This is a strong indication that our solution is correct. However, to further strengthen our confidence, it's beneficial to test another value within the solution set. Let's choose a value closer to the boundary value of 46, such as r = 40. Substituting r = 40 into the original inequality, we get:

${ \frac{40}{2} - 4 < 19 }$

Simplifying the left side of the inequality:

${ 20 - 4 < 19 }$

${ 16 < 19 }$

Again, the inequality 16 < 19 is true, further validating our solution. These two tests provide substantial evidence that the solution r < 46 is correct. To complete the verification process, it's also prudent to test a value that is not within the solution set. Let's choose a value greater than 46, such as r = 50. Substituting r = 50 into the original inequality, we get:

${ \frac{50}{2} - 4 < 19 }$

Simplifying the left side of the inequality:

${ 25 - 4 < 19 }$

${ 21 < 19 }$

The inequality 21 < 19 is false. This confirms that r = 50, a value outside our solution set, does not satisfy the original inequality. This final test provides conclusive evidence that our solution r < 46 is indeed the correct solution to the inequality ${\frac{r}{2}-4 < 19}$. The process of verifying the solution is an indispensable step in solving any mathematical problem, particularly inequalities. It helps to identify potential errors and ensures the accuracy of the final result. By substituting values from within and outside the solution set, we can gain confidence in the correctness of our answer. In the case of inequalities, verification is especially crucial because the solution is a range of values rather than a single value. Testing multiple values within the range provides a robust confirmation of the solution's validity. In conclusion, the verification step not only confirms the correctness of the solution but also reinforces the understanding of the inequality and its solution set.

Conclusion: Summarizing the Solution

In this comprehensive guide, we have meticulously dissected the process of solving the inequality ${\frac{r}{2}-4 < 19}$. We embarked on this journey by first understanding the fundamental principles of inequalities and the importance of isolating the variable to determine the solution set. The solution set, unlike an equation's single value solution, represents a range of values that satisfy the given inequality. This understanding forms the bedrock for effectively manipulating inequalities and interpreting their solutions.

Our step-by-step approach began with isolating the variable term. By adding 4 to both sides of the inequality, we successfully eliminated the constant term (-4) from the left side, simplifying the inequality to ${\frac{r}{2} < 23}$. This step demonstrated the application of the addition property of inequality, a cornerstone principle that allows us to add the same value to both sides without altering the inequality's direction. The importance of maintaining balance and performing identical operations on both sides cannot be overstated, as it ensures the integrity of the inequality throughout the solution process. Next, we focused on solving for r. To eliminate the denominator 2, we multiplied both sides of the inequality by 2. This crucial step invoked the multiplication property of inequality, which dictates that multiplying both sides by a positive value preserves the direction of the inequality. This led us to the solution r < 46, which signifies that any value of r strictly less than 46 will satisfy the original inequality. The ability to apply the multiplication property with confidence, while remembering the critical caveat of sign-flipping when multiplying by a negative number, is a key skill in inequality manipulation. With the solution in hand, we proceeded to the vital step of verification. This step serves as a safeguard against potential errors and reinforces our understanding of the solution set. By substituting values from within and outside the solution set back into the original inequality, we rigorously tested the validity of our answer. The successful verification process, involving multiple test values, provided conclusive evidence that r < 46 is indeed the correct solution.

The solution r < 46 represents a range of values extending from negative infinity up to, but not including, 46. This can be visually represented on a number line as an open interval, highlighting the infinite nature of the solution set. The process of solving inequalities, as demonstrated in this guide, requires a systematic and methodical approach. Each step must be carefully executed, with a clear understanding of the underlying principles. The ability to manipulate inequalities with confidence is a valuable skill in mathematics and its applications. In conclusion, we have successfully solved the inequality ${\frac{r}{2}-4 < 19}$, arriving at the solution r < 46. This solution represents a range of values that satisfy the inequality, and its correctness has been rigorously verified. By mastering the steps and principles outlined in this guide, you will be well-equipped to tackle a wide range of inequality problems.