Solving The Inequality 8x - 6 > 12 + 2x Step By Step
In this article, we will delve into the process of solving the inequality 8x - 6 > 12 + 2x. Inequalities are mathematical statements that compare two expressions using symbols like '>', '<', '≥', or '≤'. Solving an inequality involves finding the range of values for the variable that makes the inequality true. This is a fundamental concept in algebra and is crucial for various mathematical applications. Understanding how to solve inequalities is essential for students, educators, and anyone who uses mathematical concepts in their daily lives. We will explore each step methodically to ensure a clear understanding of the solution. By the end of this article, you will be equipped with the knowledge to tackle similar inequality problems with confidence. This guide not only provides the solution but also explains the underlying principles, making it an invaluable resource for learning and review. So, let's embark on this mathematical journey and unravel the solution to the inequality 8x - 6 > 12 + 2x. Understanding the steps involved in solving this inequality will not only help you answer this specific question but also provide a solid foundation for handling more complex inequalities in the future. The step-by-step approach we will use will break down the problem into manageable parts, making it easier to grasp the concepts and apply them effectively. Whether you are a student preparing for an exam or someone looking to refresh your algebra skills, this article is designed to provide a comprehensive and accessible explanation.
Understanding the Inequality
Before we dive into the solution, let's break down the given inequality 8x - 6 > 12 + 2x. An inequality is a mathematical statement that compares two expressions using inequality symbols. In this case, the '>' symbol indicates 'greater than'. Our goal is to find the values of 'x' that satisfy this inequality. This means we want to find all the 'x' values for which 8x - 6 is greater than 12 + 2x. To achieve this, we need to isolate 'x' on one side of the inequality. This process involves several algebraic manipulations, each designed to simplify the inequality while preserving its solution set. Understanding the meaning of the inequality is crucial because it guides the steps we take to solve it. For instance, we must be mindful of how operations like multiplying or dividing by a negative number can affect the direction of the inequality. Moreover, recognizing the structure of the inequality helps in planning the most efficient way to isolate 'x'. The expression 8x - 6 represents a linear function, and so does 12 + 2x. The inequality essentially asks for what values of 'x' the first function lies above the second. This perspective can be particularly helpful in visualizing the solution graphically. By understanding the inequality conceptually, we can approach the algebraic manipulations with greater clarity and purpose, ensuring we arrive at the correct solution. Let's move forward with the steps to isolate 'x' and find the solution set.
Step-by-Step Solution
Now, let's walk through the step-by-step solution of the inequality 8x - 6 > 12 + 2x. This process involves several algebraic manipulations to isolate 'x' on one side of the inequality. Each step is crucial to maintain the integrity of the inequality and ensure we arrive at the correct solution. We will start by moving all terms containing 'x' to one side and constant terms to the other. This is a standard technique in solving inequalities and equations. It allows us to simplify the expression and bring like terms together. The steps include addition, subtraction, and potentially division or multiplication. However, it's essential to remember that multiplying or dividing by a negative number reverses the direction of the inequality sign. This is a critical rule to keep in mind to avoid errors. The ultimate goal is to express the inequality in the form 'x > some value' or 'x < some value', which directly gives us the solution set. Each step in the process will be explained in detail to ensure clarity and understanding. By following these steps carefully, you will be able to solve this inequality and similar problems with confidence. Remember, the key is to perform the same operations on both sides of the inequality to maintain balance and preserve the solution set. Let’s begin with the first step to isolate the 'x' terms.
Step 1: Subtract 2x from Both Sides
The first step in solving the inequality 8x - 6 > 12 + 2x is to subtract 2x from both sides. This operation helps us gather all the terms containing 'x' on one side of the inequality. Subtracting the same quantity from both sides maintains the balance of the inequality, ensuring that the solution set remains unchanged. This is a fundamental principle in solving both equations and inequalities. By subtracting 2x, we eliminate the 'x' term from the right side of the inequality, which moves us closer to isolating 'x' on the left side. The resulting inequality will be simpler and easier to manipulate. This step is a standard algebraic technique and is crucial for solving various types of inequalities. It is also a good example of how inverse operations are used to simplify expressions. Understanding why we perform this step is just as important as knowing how to do it. It sets the stage for further simplification and ultimately leads us to the solution. The resulting inequality after this step will be a crucial stepping stone in the solution process. Let's perform the subtraction and see what we get. After subtracting 2x from both sides, we are one step closer to isolating 'x'. This meticulous approach ensures accuracy and a thorough understanding of the solution process.
8x - 6 - 2x > 12 + 2x - 2x
This simplifies to:
6x - 6 > 12
Step 2: Add 6 to Both Sides
Having simplified the inequality to 6x - 6 > 12, the next step is to isolate the term with 'x' further. To achieve this, we add 6 to both sides of the inequality. This operation eliminates the constant term on the left side, bringing us closer to having 'x' by itself. Just like in the previous step, adding the same value to both sides preserves the inequality. This is a crucial concept in maintaining the solution set. This step is another application of using inverse operations to simplify expressions. Adding 6 is the inverse operation of subtracting 6, which allows us to cancel out the constant term on the left side. This technique is widely used in solving equations and inequalities, making it a fundamental skill in algebra. Understanding why we add 6 is as important as knowing how to add it. It is a strategic move that simplifies the inequality and moves us closer to the final solution. By adding 6 to both sides, we create a more straightforward inequality that can be easily solved in the next step. This systematic approach is essential for solving mathematical problems accurately and efficiently. Let's proceed with the addition and observe the result. The resulting inequality will be significantly simpler and easier to solve for 'x'. This methodical progression is key to mastering algebraic manipulations.
6x - 6 + 6 > 12 + 6
This simplifies to:
6x > 18
Step 3: Divide Both Sides by 6
We've now simplified the inequality to 6x > 18. To finally isolate 'x', the next step is to divide both sides of the inequality by 6. This is the last algebraic manipulation required to solve for 'x'. Dividing both sides by the same positive number maintains the direction of the inequality, which is a critical rule in solving inequalities. This step is the final application of inverse operations. Division is the inverse operation of multiplication, and dividing by 6 cancels out the multiplication by 6 on the left side, leaving 'x' isolated. This technique is fundamental in algebra and is used extensively in solving various types of equations and inequalities. Understanding why we divide by 6 is crucial. It's the logical step to isolate 'x' and find the values that satisfy the original inequality. By dividing both sides by 6, we arrive at a simple inequality that directly tells us the solution set for 'x'. This methodical approach ensures accuracy and a thorough understanding of the solution process. Let's perform the division and see the final result. The resulting inequality will give us the range of values for 'x' that satisfy the original inequality. This is the culmination of our step-by-step solution process, showcasing the power of algebraic manipulation.
(6x) / 6 > 18 / 6
This simplifies to:
x > 3
Solution Set
After performing all the necessary steps, we have arrived at the solution: x > 3. This inequality tells us that the solution set includes all values of 'x' that are greater than 3. In other words, any number larger than 3 will satisfy the original inequality 8x - 6 > 12 + 2x. Understanding the solution set is crucial because it provides a complete picture of all possible values of 'x' that make the inequality true. The solution set can be represented in various ways, including graphically on a number line, where we would shade the region to the right of 3 (not including 3 itself). It can also be expressed in interval notation as (3, ∞), which means all real numbers greater than 3. The inequality x > 3 is a concise way of summarizing this infinite set of solutions. To verify the solution, we can test a value greater than 3 in the original inequality. For example, let's try x = 4: 8(4) - 6 = 26 and 12 + 2(4) = 20. Since 26 > 20, the inequality holds true. This confirms that our solution set is correct. The solution set is a fundamental concept in algebra and is essential for understanding the behavior of inequalities. It provides a clear and complete answer to the problem, allowing us to identify all the values that satisfy the given conditions. Let's now consider the answer choices provided in the question to determine which value of 'x' is in the solution set.
Determining the Correct Answer
Now that we have determined the solution set x > 3, we can look at the given answer choices to find the value of 'x' that satisfies this inequality. The answer choices are typically presented as multiple-choice options, and our task is to identify the one that falls within the solution set. This involves comparing each answer choice to the inequality x > 3. Any value of 'x' that is greater than 3 will be a correct answer. The provided options are:
A. -1
B. 0
C. 3
D. 5
Let's analyze each option:
- Option A: -1 is not greater than 3.
- Option B: 0 is not greater than 3.
- Option C: 3 is not greater than 3 (it is equal to 3, but the inequality is strictly 'greater than').
- Option D: 5 is greater than 3.
Based on this analysis, only option D, 5, satisfies the inequality x > 3. Therefore, the correct answer is D. This process of checking answer choices against the solution set is a standard technique in solving multiple-choice questions involving inequalities. It ensures that the selected answer is not only mathematically correct but also satisfies the specific conditions of the problem. Understanding how to interpret the solution set and apply it to the answer choices is a crucial skill in algebra and problem-solving. This step solidifies our understanding of the solution and confirms that we have correctly answered the question. Let's summarize our findings and conclude the solution.
Conclusion
In conclusion, we have successfully solved the inequality 8x - 6 > 12 + 2x and determined that the solution set is x > 3. This means that any value of 'x' greater than 3 will satisfy the original inequality. We arrived at this solution by following a step-by-step approach, which included subtracting 2x from both sides, adding 6 to both sides, and finally dividing both sides by 6. Each step was performed meticulously to ensure the integrity of the inequality and the accuracy of the solution. We then analyzed the given answer choices and identified that only 5 is greater than 3, making it the correct answer. This process highlights the importance of understanding algebraic manipulations and how they affect inequalities. It also demonstrates the significance of interpreting the solution set correctly and applying it to the given problem context. Inequalities are a fundamental concept in mathematics, and mastering their solution is essential for various applications. This article has provided a comprehensive guide to solving this specific inequality, but the principles and techniques discussed can be applied to a wide range of similar problems. The step-by-step approach, the emphasis on maintaining the integrity of the inequality, and the careful interpretation of the solution set are all key elements in solving inequalities successfully. By understanding these elements, you can confidently tackle inequality problems and enhance your mathematical skills. Remember, practice is crucial for mastering these concepts, so continue to solve similar problems to reinforce your understanding.