Solving $-\frac{3}{4}z - 18$ With Given Conditions Find The Solution For S

Introduction

In this comprehensive article, we will delve into the process of solving the inequality $-\frac{3}{4}z - 18$ for $z$, while considering the given conditions $z \leq 72$, $z \leq 22$, $z \geq -72$, and $z \geq -22$. Inequalities, a fundamental concept in mathematics, play a crucial role in various fields, including algebra, calculus, and real-world problem-solving. Understanding how to manipulate and solve inequalities is essential for building a strong foundation in mathematical reasoning. This article aims to provide a step-by-step guide to solving the given inequality, along with clear explanations and illustrative examples. We will explore the properties of inequalities, the rules for performing operations on them, and how to interpret the solutions in the context of the given conditions. By the end of this article, you will have a solid understanding of how to solve linear inequalities and how to apply this knowledge to similar problems. This exploration will not only enhance your mathematical skills but also develop your ability to think critically and solve problems logically. We will break down each step, ensuring clarity and understanding at every stage of the solution. This approach will help you grasp the underlying concepts and apply them effectively in various mathematical scenarios. Join us as we navigate through the intricacies of this inequality and uncover the solutions it holds.

Step 1: Isolate the Variable Term

To begin solving the inequality $-\frac{3}{4}z - 18$, our first objective is to isolate the term containing the variable, which in this case is $-\frac{3}{4}z$. This involves eliminating the constant term, $-18$, from the left side of the inequality. To achieve this, we apply the addition property of inequality, which states that adding the same quantity to both sides of an inequality preserves the inequality's direction. Therefore, we add 18 to both sides of the inequality:

$-\frac{3}{4}z - 18 + 18 \geq s + 18$

This simplifies to:

$-\frac{3}{4}z \geq s + 18$

Isolating the variable term is a crucial step in solving inequalities. It sets the stage for the next operation, which involves dealing with the coefficient of the variable. By adding 18 to both sides, we have effectively moved the constant term to the right side, leaving the term with $z$ isolated on the left. This process is analogous to solving equations, where the goal is to isolate the variable to determine its value. In the context of inequalities, however, we are not looking for a single value but rather a range of values that satisfy the inequality. This initial step is fundamental to simplifying the inequality and bringing us closer to finding the solution set. The careful application of the addition property ensures that the inequality remains balanced and that we are progressing towards the correct solution. This meticulous approach is vital in mathematics, as even a small error can lead to an incorrect answer. By focusing on the principles of algebraic manipulation, we are building a strong foundation for solving more complex inequalities in the future.

Step 2: Multiply by the Reciprocal

Having isolated the term $-\frac{3}{4}z$, the next step is to eliminate the coefficient $-\frac{3}{4}$. To do this, we multiply both sides of the inequality by the reciprocal of $-\frac{3}{4}$, which is $-\frac{4}{3}$. However, a crucial rule in dealing with inequalities is that multiplying or dividing both sides by a negative number reverses the direction of the inequality sign. Therefore, the $\geq$ sign will become a $\leq$ sign. Multiplying both sides by $-\frac{4}{3}$, we get:

$(-\frac{4}{3})(-\frac{3}{4}z) \leq (s + 18)(-\frac{4}{3})$

This simplifies to:

$z \leq -\frac{4}{3}(s + 18)$

Multiplying by the reciprocal is a powerful technique in solving inequalities and equations. It allows us to isolate the variable completely, giving us a clear understanding of the solution set. The critical aspect to remember here is the sign reversal when multiplying or dividing by a negative number. This is a fundamental rule in inequality manipulation and is essential for obtaining the correct solution. In this step, we have not only eliminated the coefficient but also accounted for the sign change, ensuring that our inequality remains accurate. This careful attention to detail is what makes mathematical problem-solving precise and reliable. The resulting inequality, $z \leq -\frac{4}{3}(s + 18)$, provides a direct relationship between $z$ and $s$, allowing us to explore the possible values of $z$ under different conditions. This step is a testament to the importance of understanding the properties of inequalities and applying them correctly. By mastering these techniques, we can confidently tackle a wide range of mathematical problems involving inequalities.

Step 3: Apply the Given Conditions

Now that we have the inequality $z \leq -\frac{4}{3}(s + 18)$, we need to consider the given conditions: $z \leq 72$, $z \leq 22$, $z \geq -72$, and $z \geq -22$. These conditions provide additional constraints on the possible values of $z$. To determine the solution set, we must analyze how these conditions interact with our derived inequality. Let's consider each condition individually and then combine them to find the most restrictive range for $z$.

First, the conditions $z \leq 72$ and $z \leq 22$ imply that $z$ must be less than or equal to both 72 and 22. The more restrictive of these two conditions is $z \leq 22$, as any value of $z$ that satisfies $z \leq 22$ will also satisfy $z \leq 72$. Similarly, the conditions $z \geq -72$ and $z \geq -22$ imply that $z$ must be greater than or equal to both -72 and -22. The more restrictive of these two conditions is $z \geq -22$, as any value of $z$ that satisfies $z \geq -22$ will also satisfy $z \geq -72$. Combining these more restrictive conditions, we have $-22 \leq z \leq 22$. This compound inequality defines the range of possible values for $z$ based on the given constraints.

Applying these conditions is a crucial step in solving the problem. It demonstrates the importance of considering all available information and how different constraints can interact to narrow down the solution set. In this case, we have effectively reduced the possible values of $z$ to a specific range, which is a significant step towards finding the final solution. This process highlights the mathematical skill of logical deduction, where we use given information to derive further conclusions. The ability to analyze and synthesize different conditions is essential in many areas of mathematics and problem-solving. By carefully considering each constraint, we ensure that our solution is not only mathematically correct but also logically sound. This comprehensive approach is what distinguishes effective problem solvers from those who rely solely on rote memorization.

Step 4: Substitute the Boundary Conditions

To further refine our solution, we will substitute the boundary conditions $z = -22$ and $z = 22$ into the inequality $z \leq -\frac{4}{3}(s + 18)$ to find the corresponding values of $s$. This will help us determine the range of $s$ that satisfies the given conditions.

First, let's substitute $z = -22$:

$-22 \leq -\frac{4}{3}(s + 18)$

To solve for $s$, we multiply both sides by $-\frac{3}{4}$, remembering to reverse the inequality sign:

$-22(-\frac{3}{4}) \geq s + 18$

This simplifies to:

$\frac{33}{2} \geq s + 18$

Subtracting 18 from both sides, we get:

$\frac{33}{2} - 18 \geq s$

$\frac{33}{2} - \frac{36}{2} \geq s$

$-\frac{3}{2} \geq s$

So, $s \leq -\frac{3}{2}$.

Next, let's substitute $z = 22$:

$22 \leq -\frac{4}{3}(s + 18)$

Multiplying both sides by $-\frac{3}{4}$ and reversing the inequality sign:

$22(-\frac{3}{4}) \geq s + 18$

This simplifies to:

$-\frac{33}{2} \geq s + 18$

Subtracting 18 from both sides, we get:

$-\frac{33}{2} - 18 \geq s$

$-\frac{33}{2} - \frac{36}{2} \geq s$

$-\frac{69}{2} \geq s$

So, $s \leq -\frac{69}{2}$.

Substituting the boundary conditions allows us to connect the range of $z$ to the possible values of $s$. This step is crucial for understanding how the different variables in the inequality relate to each other. By finding the values of $s$ that correspond to the extreme values of $z$, we can establish the boundaries of the solution set for $s$. This process demonstrates the mathematical concept of mapping, where we transform the conditions on one variable to conditions on another. The calculations involved in this step require careful attention to detail and a solid understanding of algebraic manipulation. By performing these substitutions accurately, we ensure that our final solution is both mathematically sound and consistent with the given conditions. This step highlights the interconnectedness of different parts of a mathematical problem and the importance of considering these connections when seeking a solution.

Step 5: Determine the Solution Set for s

From the previous step, we found two conditions for $s$: $s \leq -\frac{3}{2}$ and $s \leq -\frac{69}{2}$. To determine the overall solution set for $s$, we need to consider both conditions and identify the most restrictive one. In this case, $s \leq -\frac{69}{2}$ is the more restrictive condition because any value of $s$ that satisfies $s \leq -\frac{69}{2}$ will also satisfy $s \leq -\frac{3}{2}$. Therefore, the solution set for $s$ is $s \leq -\frac{69}{2}$.

This solution set tells us that $s$ can take on any value less than or equal to $-\frac{69}{2}$, which is -34.5. This is a specific range of values that satisfies the original inequality and the given conditions. Determining the solution set is the final goal in solving any inequality problem. It provides a clear and concise answer that can be easily interpreted and applied in various contexts. The process of identifying the most restrictive condition highlights the mathematical skill of logical reasoning and the ability to compare different constraints. By carefully analyzing the conditions, we can arrive at a precise solution that accurately reflects the problem's requirements. This step demonstrates the importance of synthesis in mathematical problem-solving, where we combine different pieces of information to form a coherent and complete answer. The solution set $s \leq -\frac{69}{2}$ represents the culmination of our step-by-step analysis and provides a definitive answer to the problem.

Conclusion

In this article, we have thoroughly explored the process of solving the inequality $-\frac{3}{4}z - 18$ for $z$, considering the conditions $z \leq 72$, $z \leq 22$, $z \geq -72$, and $z \geq -22$. By systematically isolating the variable, applying the given conditions, and substituting boundary values, we determined that the solution set for $s$ is $s \leq -\frac{69}{2}$. This comprehensive approach highlights the importance of understanding the properties of inequalities and applying them correctly. Solving inequalities is a fundamental skill in mathematics, with applications in various fields, including physics, engineering, and economics. The ability to manipulate inequalities and interpret their solutions is essential for problem-solving and decision-making. This article has provided a detailed guide to solving a specific inequality, but the techniques and principles discussed can be applied to a wide range of similar problems. By mastering these skills, you can enhance your mathematical proficiency and confidently tackle complex problems. The step-by-step approach outlined in this article emphasizes the importance of clarity, accuracy, and logical reasoning in mathematical problem-solving. By following these principles, you can effectively solve inequalities and gain a deeper understanding of mathematical concepts. The solution we have arrived at is not just a numerical answer; it is a result of a careful and methodical process that demonstrates the power of mathematical thinking.