Solving The Inequality 8x - 6 > 12 + 2x Step By Step

What value of x is in the solution set of the inequality 8x - 6 > 12 + 2x? Explain the steps to solve the inequality.

In this article, we will delve into the process of solving the inequality 8x - 6 > 12 + 2x. This type of problem falls under the domain of algebraic inequalities, a fundamental topic in mathematics. Understanding how to solve inequalities is crucial for various mathematical applications and problem-solving scenarios. We will break down the steps involved in finding the solution set and identify the value of 'x' that satisfies the given inequality. This exploration will not only provide the answer to the specific question but also enhance your understanding of inequality manipulation and solution interpretation. Our primary focus will be on clarity and thoroughness, ensuring that each step is explained in detail, making it easier for learners of all levels to grasp the concepts involved. By the end of this guide, you will be equipped with the knowledge and skills to tackle similar inequality problems with confidence. So, let's embark on this mathematical journey together and unravel the solution to the inequality 8x - 6 > 12 + 2x.

Before we dive into solving the specific inequality, it's essential to understand what inequalities are and how they differ from equations. In mathematics, an inequality is a statement that compares two expressions using inequality symbols such as '>', '<', '≥', or '≤'. These symbols represent 'greater than', 'less than', 'greater than or equal to', and 'less than or equal to', respectively. Unlike equations, which state that two expressions are equal, inequalities indicate a range of possible values that satisfy the given condition. This range of values is known as the solution set. When solving inequalities, our goal is to isolate the variable (in this case, 'x') on one side of the inequality symbol, just like we do with equations. However, there's a crucial difference: multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality symbol. This is a key rule to remember to ensure accurate solutions. Inequalities are used extensively in various fields, including optimization problems, economics, and physics, to model situations where exact equality is not required or possible. The ability to solve and interpret inequalities is a fundamental skill in mathematics and its applications.

Let's now embark on a step-by-step journey to solve the inequality 8x - 6 > 12 + 2x. Our goal is to isolate the variable 'x' on one side of the inequality to determine the range of values that satisfy the condition. The process involves several algebraic manipulations, each designed to simplify the inequality while preserving its truth. Firstly, we need to gather all the 'x' terms on one side and the constant terms on the other. This is achieved by performing addition or subtraction operations on both sides of the inequality. Secondly, once the 'x' terms are grouped, we simplify further by combining like terms. This step reduces the inequality to a more manageable form. Thirdly, to finally isolate 'x', we may need to perform division or multiplication operations. It's crucial to remember that multiplying or dividing by a negative number requires flipping the inequality sign. Finally, after these steps, we arrive at the solution set for 'x', which represents all values that make the original inequality true. This methodical approach ensures accuracy and clarity in solving inequalities. Now, let's apply these steps to our specific problem.

To solve the inequality 8x - 6 > 12 + 2x, we will follow the steps outlined earlier, ensuring each step is clearly explained for better understanding. The first step involves moving all the 'x' terms to one side of the inequality. To do this, we subtract '2x' from both sides of the inequality: 8x - 6 - 2x > 12 + 2x - 2x. This simplifies to 6x - 6 > 12. Next, we need to isolate the 'x' term further by moving the constant term to the other side. We add 6 to both sides of the inequality: 6x - 6 + 6 > 12 + 6. This simplifies to 6x > 18. Now, to finally isolate 'x', we divide both sides of the inequality by 6: (6x) / 6 > 18 / 6. This gives us x > 3. This is the solution set for the inequality, indicating that any value of 'x' greater than 3 will satisfy the original inequality. This step-by-step approach ensures that we maintain the balance of the inequality while simplifying it to find the solution.

Now that we have determined the solution set for the inequality 8x - 6 > 12 + 2x as x > 3, the next step is to identify which of the given options falls within this range. The options provided are A. -1, B. 0, C. 3, and D. 5. To find the correct value, we need to check each option against our solution set. Option A, -1, is not greater than 3. Option B, 0, is also not greater than 3. Option C, 3, is not greater than 3 (it is equal to 3, but our solution requires 'x' to be strictly greater than 3). Option D, 5, is greater than 3. Therefore, the value of 'x' that is in the solution set of the inequality 8x - 6 > 12 + 2x is 5. This process of checking each option against the solution set is a crucial step in ensuring the correct answer is selected. By systematically evaluating each option, we can confidently identify the value that satisfies the inequality.

In conclusion, we have successfully solved the inequality 8x - 6 > 12 + 2x and identified the value of 'x' that belongs to the solution set. We began by understanding the basics of inequalities and how they differ from equations. Then, we systematically solved the inequality by isolating the variable 'x', step by step, ensuring that each operation preserved the integrity of the inequality. We arrived at the solution set x > 3, which means any value of 'x' greater than 3 will satisfy the original inequality. Finally, by comparing the given options to our solution set, we determined that the correct value is 5. This exercise demonstrates the importance of understanding algebraic manipulations and the rules governing inequalities. The ability to solve inequalities is a fundamental skill in mathematics, with applications in various fields. By mastering these concepts, you can confidently tackle a wide range of mathematical problems and real-world scenarios.