Solving Inequalities A Step-by-Step Guide To 8z + 3 - 2z < 51
Inequalities are a fundamental concept in mathematics, representing relationships where two values are not necessarily equal. Unlike equations that use an equals sign (=), inequalities use symbols such as less than (<), greater than (>), less than or equal to (≤), and greater than or equal to (≥) to express the relative order of values. In this article, we will focus on solving a specific inequality: 8z + 3 - 2z < 51. This example will serve as a comprehensive guide to understanding and applying the principles of inequality manipulation. Before we dive into the solution, let's briefly review the basic properties of inequalities and how they differ from equations.
When solving inequalities, the goal is the same as solving equations: to isolate the variable on one side of the inequality. However, there's a crucial difference to keep in mind: when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. This is because multiplying or dividing by a negative number changes the direction of the relationship between the values. For instance, if 2 < 4, multiplying both sides by -1 gives -2 > -4. This property is essential for solving inequalities correctly, and we'll see it in action as we work through our example. The ability to solve inequalities is vital in various fields, including algebra, calculus, and real-world applications where constraints and ranges are important. Understanding the nuances of inequalities allows for a more comprehensive approach to problem-solving, especially when dealing with situations that do not have a single, fixed solution but rather a range of possible solutions. Now, let's proceed to solve the given inequality step by step, ensuring a clear and thorough understanding of each operation.
Step 1: Simplify the Inequality
The first crucial step in solving the inequality 8z + 3 - 2z < 51 is to simplify it by combining like terms. In this case, we have two terms involving the variable 'z': 8z and -2z. Combining these terms makes the inequality easier to manage and reduces the complexity of the subsequent steps. By adding 8z and -2z, we get 6z. So, the inequality now becomes 6z + 3 < 51. This simplification is a fundamental algebraic technique that is used to make equations and inequalities more manageable. It allows us to consolidate terms and focus on the core relationship between the variable and the constants.
Simplification is not just a matter of reducing the number of terms; it also helps in visualizing the problem more clearly. By combining like terms, we reduce the chances of making errors in the following steps. For example, dealing with 6z instead of 8z - 2z is less prone to mistakes. This step is particularly important in more complex inequalities where multiple simplifications might be necessary. Now that we have simplified the inequality to 6z + 3 < 51, we can move on to the next step, which involves isolating the term with the variable on one side of the inequality. This will bring us closer to solving for 'z' and determining the range of values that satisfy the inequality. Remember, the goal is to get 'z' by itself, and simplification is the first key to achieving that goal.
Step 2: Isolate the Variable Term
After simplifying the inequality to 6z + 3 < 51, the next step involves isolating the variable term. To do this, we need to eliminate the constant term (+3) from the left side of the inequality. We can achieve this by subtracting 3 from both sides of the inequality. This operation maintains the balance of the inequality, just like it does in equations. Subtracting 3 from both sides gives us 6z + 3 - 3 < 51 - 3, which simplifies to 6z < 48. This step is critical because it brings us closer to isolating the variable 'z', which is the ultimate goal in solving the inequality.
The principle behind this step is based on the properties of inequalities, which state that you can add or subtract the same value from both sides without changing the direction of the inequality. This is a fundamental concept that applies to all types of inequalities. By subtracting 3, we effectively move the constant term to the right side of the inequality, leaving the term with 'z' alone on the left side. This isolation is a crucial step towards determining the possible values of 'z' that satisfy the original inequality. Now that we have 6z < 48, we can proceed to the final step, which involves dividing both sides by the coefficient of 'z' to find the solution. This next step will give us the range of values for 'z' that make the inequality true.
Step 3: Solve for the Variable
Now that we have the inequality in the form 6z < 48, the final step is to solve for 'z'. To do this, we need to isolate 'z' completely by dividing both sides of the inequality by the coefficient of 'z', which is 6. When we divide both sides by 6, we get (6z)/6 < 48/6. This simplifies to z < 8. This is the solution to the inequality. It tells us that any value of 'z' that is less than 8 will satisfy the original inequality 8z + 3 - 2z < 51.
It's crucial to remember that when we divide or multiply an inequality by a negative number, we need to reverse the inequality sign. However, in this case, we are dividing by a positive number (6), so we do not need to reverse the sign. The solution z < 8 represents a range of values, not just a single value. This is a key difference between inequalities and equations. While an equation typically has one or a few specific solutions, an inequality has a range of solutions. In this instance, any number less than 8 is a valid solution. This includes numbers like 7, 0, -5, and so on. Understanding this concept of a range of solutions is fundamental to working with inequalities. We have now successfully solved the inequality and found the solution set. In the next section, we will verify our solution and discuss how to represent it graphically.
Verification of the Solution
To ensure the accuracy of our solution, it's essential to verify it. We found that the solution to the inequality 8z + 3 - 2z < 51 is z < 8. To verify this, we can substitute a value less than 8 into the original inequality and check if it holds true. Let's choose z = 7 as our test value since it is clearly less than 8. Substituting z = 7 into the original inequality, we get 8(7) + 3 - 2(7) < 51. This simplifies to 56 + 3 - 14 < 51, which further simplifies to 45 < 51. Since 45 is indeed less than 51, our solution is verified for z = 7. This process confirms that our solution is likely correct, but it's always a good idea to test another value to be even more certain.
Let's try another value, say z = 0, which is also less than 8. Substituting z = 0 into the original inequality gives us 8(0) + 3 - 2(0) < 51. This simplifies to 0 + 3 - 0 < 51, which is 3 < 51. This is also true, further reinforcing the validity of our solution. Verification is a crucial step in problem-solving, as it helps to catch any potential errors made during the process. By substituting values that fall within the solution range, we can confirm that our algebraic manipulations were correct and that the solution we obtained accurately represents the set of values that satisfy the inequality. Now that we have verified our solution algebraically, let's discuss how to represent it graphically, which provides a visual understanding of the solution set.
Graphical Representation
Visualizing the solution to an inequality is often best achieved through a graphical representation on a number line. For the inequality z < 8, we represent all values of z that are less than 8. To do this on a number line, we draw a line and mark the number 8. Since the inequality is strictly less than (z < 8), we use an open circle at 8 to indicate that 8 itself is not included in the solution set. Then, we draw an arrow extending to the left from the open circle, indicating that all values to the left of 8 (i.e., all numbers less than 8) are solutions to the inequality. This visual representation provides a clear understanding of the range of values that satisfy the inequality.
The graphical representation is particularly useful for understanding inequalities because it allows us to see the entire solution set at a glance. For example, if the inequality were z ≤ 8, we would use a closed circle at 8 to indicate that 8 is included in the solution set. The arrow extending to the left would still indicate that all values less than 8 are solutions. Similarly, for inequalities involving “greater than” or “greater than or equal to,” the arrow would extend to the right. The number line representation is a valuable tool for both solving and interpreting inequalities, especially when dealing with more complex scenarios or systems of inequalities. It helps to connect the algebraic solution with a visual understanding, making the concept more intuitive. Now that we have both algebraically verified and graphically represented the solution, we have a comprehensive understanding of the inequality and its solution set.
In this article, we have comprehensively addressed the process of solving the inequality 8z + 3 - 2z < 51. We began by simplifying the inequality through combining like terms, which gave us 6z + 3 < 51. Then, we isolated the variable term by subtracting 3 from both sides, resulting in 6z < 48. Finally, we solved for 'z' by dividing both sides by 6, which led us to the solution z < 8. This solution indicates that any value of 'z' that is less than 8 will satisfy the original inequality. We then verified our solution by substituting values less than 8 into the original inequality and confirming that the inequality held true. This step is crucial to ensure the accuracy of our algebraic manipulations.
Furthermore, we explored the graphical representation of the solution on a number line. By using an open circle at 8 and an arrow extending to the left, we visually depicted all values less than 8 as the solution set. This graphical representation provides an intuitive understanding of the range of solutions for the inequality. The final answer to the inequality 8z + 3 - 2z < 51 is z < 8. This means that the correct option is A) z < 8. Understanding how to solve inequalities is a fundamental skill in algebra and has applications in various fields of mathematics and real-world problem-solving. By following these step-by-step procedures and verifying the solution, one can confidently solve a wide range of inequalities. Remember, the key is to simplify, isolate the variable, and apply the rules of inequalities correctly. With practice, solving inequalities becomes a straightforward and essential skill in mathematical analysis.
Final Answer: A) z < 8
