Solve The Inequality 8z + 3 - 2z < 51. Options: A. Z < 8, B. Z < 5.4, C. Z < 9, D. Z < 14.

In the world of mathematics, inequalities play a crucial role. They help us understand relationships between quantities that aren't necessarily equal. Mastering the art of solving inequalities is fundamental, whether you're tackling algebraic problems or real-world scenarios. This article will guide you through the process of solving a specific inequality: 8z + 3 - 2z < 51. We'll break down each step, ensuring a clear understanding of the techniques involved.

Inequalities, unlike equations, don't deal with strict equality. Instead, they express relationships where one quantity is greater than, less than, greater than or equal to, or less than or equal to another. The ability to solve inequalities is a cornerstone of algebra and calculus, with applications spanning across various fields, from economics to physics. Understanding how to manipulate inequalities and isolate variables is essential for problem-solving and decision-making in numerous contexts. This article serves as a comprehensive guide to solving the inequality 8z + 3 - 2z < 51, providing a step-by-step approach that not only yields the solution but also enhances your understanding of the underlying principles. Whether you are a student learning the basics of algebra or someone looking to refresh your mathematical skills, this guide will equip you with the knowledge and confidence to tackle similar problems. By focusing on clarity and detail, we aim to make the process of solving inequalities accessible and engaging, ensuring that you can apply these techniques effectively in your mathematical journey.

Before we dive into solving, let's dissect the inequality itself: 8z + 3 - 2z < 51. This expression tells us that the value of "8z plus 3 minus 2z" is strictly less than 51. Our goal is to find all the values of 'z' that make this statement true. To achieve this, we'll employ algebraic techniques to isolate 'z' on one side of the inequality. This process involves simplifying the expression, combining like terms, and performing operations on both sides of the inequality while maintaining its balance. Understanding the structure of the inequality is the first step towards finding its solution. The variable 'z' represents an unknown value, and the inequality sets a condition that this value must satisfy. By carefully manipulating the inequality, we can narrow down the possible values of 'z' until we arrive at the solution set. This solution set may consist of a range of values rather than a single value, which is a key difference between solving inequalities and solving equations. The '<' symbol indicates that the expression on the left-hand side must be strictly less than 51; it cannot be equal to 51. This distinction is important when interpreting the final solution, as it will define the boundary of the solution set. As we progress through the steps, we will pay close attention to how each operation affects the inequality and ensure that we maintain its validity throughout the process.

The first step in solving the inequality 8z + 3 - 2z < 51 is to simplify the expression on the left-hand side. We can combine the 'z' terms: 8z - 2z = 6z. This gives us a simplified inequality: 6z + 3 < 51. Simplifying expressions is a fundamental technique in algebra, allowing us to work with more manageable forms of equations and inequalities. By combining like terms, we reduce the complexity of the expression, making it easier to isolate the variable. In this case, combining the 'z' terms not only simplifies the inequality but also brings us closer to our goal of isolating 'z'. The simplified inequality, 6z + 3 < 51, is mathematically equivalent to the original inequality, but it is in a form that is much easier to work with. This step highlights the importance of algebraic manipulation in solving mathematical problems. By strategically simplifying the expression, we pave the way for subsequent steps that will lead us to the solution. The ability to identify and combine like terms is a crucial skill in algebra, and this step provides a clear example of its application. As we move forward, we will continue to employ algebraic techniques to further isolate 'z' and determine the range of values that satisfy the inequality.

To further isolate 'z', we need to get the term with 'z' (which is 6z) by itself on one side of the inequality. We can do this by subtracting 3 from both sides: 6z + 3 - 3 < 51 - 3. This simplifies to 6z < 48. Remember, whatever operation we perform on one side of an inequality, we must perform on the other side to maintain the balance. This principle is crucial in solving inequalities and ensures that the solution set remains unchanged. Subtracting 3 from both sides effectively cancels out the '+ 3' on the left-hand side, leaving us with the term containing 'z' isolated. This step is a classic example of using inverse operations to isolate a variable. By subtracting, we undo the addition, moving us closer to our objective. The resulting inequality, 6z < 48, is a significant step forward in our solution process. We have successfully isolated the variable term, and now we are just one step away from finding the solution set for 'z'. The importance of maintaining balance in inequalities cannot be overstated. Any operation performed on only one side would alter the relationship between the two expressions and lead to an incorrect solution. Therefore, we must always apply the same operation to both sides to preserve the validity of the inequality.

Now, to finally solve for 'z', we need to divide both sides of the inequality by 6: (6z) / 6 < 48 / 6. This gives us z < 8. This is our solution! It means that any value of 'z' that is less than 8 will satisfy the original inequality. Dividing both sides of the inequality by 6 isolates 'z' and reveals the range of values that make the inequality true. It's important to note that when dividing or multiplying both sides of an inequality by a negative number, we must flip the direction of the inequality sign. However, in this case, we are dividing by a positive number (6), so the inequality sign remains the same. The solution, z < 8, represents an infinite set of values. Any number less than 8, whether it's 7.99, 0, -1, or -100, will satisfy the original inequality. This is a key characteristic of inequalities – they often have a range of solutions rather than a single solution. Understanding this concept is crucial for interpreting the results and applying them in real-world contexts. The final step of dividing to isolate the variable highlights the power of inverse operations in solving algebraic problems. By undoing the multiplication, we reveal the solution set and complete the process of solving the inequality.

The solution to the inequality 8z + 3 - 2z < 51 is z < 8. This corresponds to option A. In conclusion, solving inequalities involves simplifying, isolating the variable term, and then solving for the variable. Remember to perform the same operations on both sides and to flip the inequality sign if you multiply or divide by a negative number. Understanding the solution z < 8 means that any value of 'z' less than 8 will make the original inequality true. This range of values represents the solution set, and it's important to recognize that inequalities often have an infinite number of solutions. The process we've outlined here is applicable to a wide range of linear inequalities, providing a solid foundation for tackling more complex problems. The key takeaways from this exercise include the importance of simplifying expressions, using inverse operations to isolate variables, and maintaining balance by performing the same operations on both sides of the inequality. Additionally, it's crucial to remember the rule about flipping the inequality sign when multiplying or dividing by a negative number. By mastering these techniques, you can confidently solve linear inequalities and apply them in various mathematical and real-world scenarios. The ability to solve inequalities is a valuable skill that extends beyond the classroom, enabling you to make informed decisions and solve problems in everyday life.