Solving Inequalities Step By Step Guide For 8z + 3 - 2z Less Than 51

This article provides a detailed, step-by-step solution to the inequality 8z + 3 - 2z < 51. Understanding how to solve inequalities is a fundamental skill in algebra, with applications ranging from simple problem-solving to more complex mathematical modeling. We will break down each step clearly, ensuring that you grasp the underlying concepts and can confidently tackle similar problems. Whether you're a student learning algebra for the first time or someone looking to refresh your skills, this guide will provide a thorough understanding of solving linear inequalities. The goal is to solve inequalities by isolating the variable. The step-by-step approach we will use involves simplifying the inequality, combining like terms, and performing operations on both sides to isolate the variable. It is crucial to understand that when you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. This is a key rule that ensures the solution set remains accurate. Furthermore, we will emphasize the importance of checking your solution to avoid common mistakes. By substituting a value from your solution set back into the original inequality, you can verify that your answer is correct. This practice not only reinforces your understanding but also helps you build confidence in your problem-solving abilities. Linear inequalities are used extensively in various fields, including economics, physics, and engineering, to model real-world constraints and optimizations. For example, in economics, inequalities might represent budget constraints or profit margins. In physics, they could describe the range of possible values for velocity or acceleration. Understanding these applications can motivate your learning and highlight the practical relevance of mastering this algebraic skill. By the end of this guide, you will not only know how to solve the specific inequality 8z + 3 - 2z < 51, but also have a solid foundation for solving a wide range of linear inequalities. This will empower you to approach more advanced mathematical topics with greater confidence and proficiency. So, let's dive in and break down the steps needed to solve this inequality effectively!

Step 1: Combine Like Terms

The first crucial step in solving the inequality is to combine like terms. We begin with the inequality: $8z + 3 - 2z < 51$. Observe that we have two terms involving the variable 'z': 8z and -2z. These are like terms because they both contain the same variable raised to the same power (in this case, z to the power of 1). To combine them, we perform the operation indicated, which is subtraction in this case. So, we subtract 2z from 8z. This process of combining like terms is a fundamental principle in algebra. It simplifies the expression and makes it easier to work with. Without this step, the equation would be more complex and challenging to solve. The ability to quickly and accurately combine like terms is essential for solving more complicated algebraic equations and inequalities. It's a skill that lays the groundwork for more advanced mathematical concepts. Remember, like terms can only be combined if they have the same variable and the same exponent. For example, 8z and -2z can be combined, but 8z and 3 cannot, because 3 is a constant term and does not have the variable 'z'. When teaching this concept, it's helpful to use visual aids or real-world examples. For instance, you could say, "If you have 8 apples and you take away 2 apples, how many apples do you have left?" This makes the abstract concept of combining like terms more concrete and relatable. This step is not only about simplifying the expression but also about organizing the equation in a way that makes the next steps more straightforward. By reducing the number of terms, we reduce the complexity of the equation, making it easier to isolate the variable and solve for it. This process mirrors real-world problem-solving strategies, where breaking down a large problem into smaller, manageable parts is often the key to finding a solution. The result of combining 8z and -2z is 6z. So, the inequality now looks like this: $6z + 3 < 51$. This simplified form is much easier to work with and sets the stage for the next steps in solving the inequality. The importance of this step cannot be overstated, as it forms the basis for all subsequent steps. By mastering the skill of combining like terms, you are building a solid foundation for more advanced algebraic concepts. This skill is not just limited to solving inequalities; it is a fundamental part of simplifying any algebraic expression. Thus, understanding and practicing this step thoroughly will greatly enhance your mathematical abilities. Now that we have combined like terms, we have a more streamlined inequality to work with. Let's move on to the next step, where we will further isolate the variable 'z'.

Step 2: Isolate the Variable Term

After combining like terms, the next crucial step is to isolate the variable term. Our inequality is now $6z + 3 < 51$. To isolate the term with 'z' (which is 6z), we need to eliminate the constant term on the same side of the inequality. In this case, the constant term is +3. To eliminate +3, we perform the inverse operation, which is subtraction. We subtract 3 from both sides of the inequality. This is a critical step because it maintains the balance of the inequality. Whatever operation we perform on one side, we must perform the same operation on the other side to ensure the inequality remains valid. This principle is similar to the rules for solving equations, where maintaining balance is essential. Understanding this concept is vital for accurately solving any algebraic equation or inequality. When you subtract 3 from the left side (6z + 3), the +3 and -3 cancel each other out, leaving us with just 6z. On the right side, we subtract 3 from 51, which gives us 48. So, the inequality becomes $6z < 48$. This step has successfully isolated the variable term on one side of the inequality, making it easier to solve for 'z'. By isolating the variable term, we are one step closer to determining the solution set for the inequality. This process of isolating terms is a fundamental technique in algebra and is used extensively in solving various types of equations and inequalities. It’s important to perform the same operation on both sides of the inequality to maintain its integrity. If we were to subtract 3 only from one side, the inequality would no longer be balanced, and the solution we obtain would be incorrect. This concept of balance is a cornerstone of algebraic manipulations. Visualizing the inequality as a balanced scale can be helpful in understanding this principle. Any operation performed on one side must be mirrored on the other side to keep the scale balanced. This analogy can make the abstract concept of algebraic manipulation more concrete and easier to grasp. In this case, subtracting 3 from both sides keeps the inequality balanced and allows us to progress toward isolating the variable. Now that we have successfully isolated the variable term, the inequality is in a much simpler form. The next step will involve isolating the variable 'z' completely, which will lead us to the solution of the inequality. Let's proceed to that step and see how we can finalize the solution.

Step 3: Solve for the Variable

Having isolated the variable term, the final step is to solve for the variable 'z'. Our inequality is now $6z < 48$. To solve for 'z', we need to isolate 'z' completely. Currently, 'z' is being multiplied by 6. To undo this multiplication, we perform the inverse operation, which is division. We divide both sides of the inequality by 6. This is a crucial step, as it directly leads us to the solution for 'z'. Just like in the previous step, it's essential to perform the same operation on both sides of the inequality to maintain balance and ensure the solution remains accurate. This principle of maintaining balance is fundamental in algebra and applies to solving equations and inequalities alike. When you divide 6z by 6, the 6s cancel each other out, leaving us with just 'z' on the left side. On the right side, we divide 48 by 6, which results in 8. Therefore, the inequality becomes $z < 8$. This is the solution to the inequality. It tells us that 'z' can be any value less than 8. This range of values represents the solution set for the inequality. Understanding what this solution means is crucial. It's not just about getting the correct number; it's about interpreting the result in the context of the original problem. In this case, 'z' can take on any value that is strictly less than 8. This means 7.99, 7, 0, -1, and so on, are all valid solutions. This concept of a solution set is different from solving equations, where we typically find a single value for the variable. In inequalities, we often find a range of values that satisfy the condition. This difference is important to grasp, as it highlights the nature of inequalities and their applications in various mathematical and real-world scenarios. The solution $z < 8$ can be represented graphically on a number line, where we would draw an open circle at 8 (to indicate that 8 is not included in the solution) and shade the line to the left, representing all values less than 8. This visual representation can be a helpful tool for understanding the solution set of an inequality. Now that we have solved for 'z', it's always a good practice to check our solution. We can do this by substituting a value from our solution set back into the original inequality. This will verify that our solution is correct. Let’s move on to the next step, where we will check our solution to ensure its accuracy.

Step 4: Verify the Solution

After obtaining a solution, it is imperative to verify the solution to ensure accuracy. Our solution to the inequality $8z + 3 - 2z < 51$ is $z < 8$. To verify this solution, we will substitute a value less than 8 into the original inequality. Choosing a simple value like z = 0 makes the calculation straightforward. Substituting z = 0 into the inequality, we get: $8(0) + 3 - 2(0) < 51$ Simplifying this, we have: $0 + 3 - 0 < 51$ $3 < 51$ This statement is true, which confirms that our solution $z < 8$ is likely correct. However, one check is not always sufficient to guarantee the correctness of the solution. It's a good practice to check with another value, especially one closer to the boundary of our solution set. Let's try z = 7. Substituting z = 7 into the original inequality, we get: $8(7) + 3 - 2(7) < 51$ $56 + 3 - 14 < 51$ $45 < 51$ This statement is also true, further reinforcing our confidence in the solution $z < 8$. The process of verifying the solution is crucial because it helps identify any potential errors made during the solving process. Mistakes can occur during any step, such as incorrectly combining like terms, not properly distributing a negative sign, or failing to reverse the inequality sign when multiplying or dividing by a negative number. By checking the solution, we can catch these errors and correct them. This step also reinforces the understanding of what the solution means. The solution $z < 8$ represents a range of values, not just a single value. By substituting values from this range back into the original inequality, we are essentially testing whether these values satisfy the inequality condition. This hands-on approach solidifies the concept of a solution set and its implications. Furthermore, verifying the solution is a valuable skill that promotes critical thinking and attention to detail. It encourages a methodical approach to problem-solving and instills the habit of double-checking work, which is essential not only in mathematics but also in various other fields. In summary, verifying the solution is an indispensable step in solving inequalities. It provides assurance that the solution is correct, helps identify and correct errors, reinforces understanding, and promotes critical thinking. Now that we have thoroughly verified our solution, we can confidently state that the solution to the inequality $8z + 3 - 2z < 51$ is indeed $z < 8$.

Final Answer

After carefully solving and verifying the inequality $8z + 3 - 2z < 51$, we have arrived at the final answer. By combining like terms, isolating the variable term, solving for the variable, and verifying our solution, we have ensured the accuracy of our result. The steps we followed are crucial for solving any linear inequality, and understanding these steps thoroughly will enable you to tackle more complex problems with confidence. Let's reiterate the solution process briefly. First, we combined the like terms 8z and -2z, resulting in 6z. This simplified the inequality to $6z + 3 < 51$. Next, we isolated the variable term by subtracting 3 from both sides, which gave us $6z < 48$. Then, we solved for 'z' by dividing both sides by 6, leading to the solution $z < 8$. Finally, we verified our solution by substituting values less than 8 back into the original inequality, confirming its correctness. The final answer, therefore, is: $z < 8$ This means that any value of 'z' that is less than 8 will satisfy the original inequality. It's important to remember that inequalities represent a range of solutions, unlike equations which typically have a single solution or a finite set of solutions. The solution $z < 8$ represents an infinite number of values, all of which are less than 8. This understanding is crucial for interpreting and applying the solution in various contexts. In practical applications, inequalities are used to model constraints and limitations. For example, in budgeting, an inequality might represent the maximum amount of money that can be spent. In manufacturing, it could represent the minimum or maximum production levels to meet demand. Understanding how to solve and interpret inequalities is therefore essential in many real-world scenarios. Moreover, the skills developed in solving inequalities are transferable to other areas of mathematics and problem-solving. The logical steps involved, such as simplifying expressions, isolating variables, and verifying solutions, are fundamental techniques that apply to a wide range of mathematical problems. By mastering these skills, you are building a strong foundation for further mathematical studies. In conclusion, the solution to the inequality $8z + 3 - 2z < 51$ is $z < 8$. This solution has been obtained through a systematic process of simplification, isolation, division, and verification. By understanding and applying these steps, you can confidently solve similar inequalities and tackle more complex mathematical challenges. The process of solving inequalities, as demonstrated in this article, is not just about finding the correct answer. It's about developing a methodical and logical approach to problem-solving. Each step in the process is designed to simplify the problem and move closer to the solution. This structured approach is a valuable skill that can be applied in various areas of life, both inside and outside the realm of mathematics. The ability to break down a complex problem into smaller, manageable steps is a key attribute of effective problem-solvers. By practicing and mastering the techniques presented in this article, you are not only improving your mathematical skills but also honing your problem-solving abilities in general. So, keep practicing, keep verifying your solutions, and keep building your confidence in tackling mathematical challenges.

Therefore, the correct answer is:

A. z < 8