Solving Inequalities A Step By Step Guide To 3x + 3 - X + (-7) > 6

In the realm of mathematics, inequalities play a crucial role in defining ranges and conditions for solutions. This article delves into the process of solving a specific inequality, 3x + 3 - x + (-7) > 6, providing a comprehensive step-by-step guide to arrive at the correct solution. Understanding inequalities is fundamental for various mathematical concepts and real-world applications. Inequalities are mathematical statements that compare two expressions using symbols like '>', '<', '≥', or '≤'. Unlike equations, which have a single solution or a finite set of solutions, inequalities often have a range of values that satisfy the given condition. This makes solving inequalities a slightly different process than solving equations, requiring careful consideration of the inequality sign and its implications. Let's embark on this mathematical journey, breaking down the problem into manageable steps and exploring the underlying principles that govern the solution.

Before diving into the specific inequality, it's essential to grasp the basic principles of solving inequalities. The goal is to isolate the variable (in this case, 'x') on one side of the inequality, just as we do with equations. However, there's a crucial difference: multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign. This is a key rule to remember and apply correctly. Think of it this way: if you have 2 < 4, multiplying both sides by -1 gives -2 and -4. Clearly, -2 is greater than -4, so the inequality sign must flip. With this understanding, we can approach the given inequality with confidence. We'll start by simplifying the expression, combining like terms, and then isolating 'x' to determine the range of values that satisfy the inequality. The process involves careful algebraic manipulation, ensuring that each step maintains the integrity of the inequality and leads us closer to the solution. So, let's get started and unravel the solution to this mathematical puzzle.

The initial step in solving any algebraic expression, whether it's an equation or an inequality, is simplification. This involves combining like terms and reducing the expression to its simplest form. In our case, the inequality is 3x + 3 - x + (-7) > 6. To simplify this, we first identify the like terms: the terms containing 'x' (3x and -x) and the constant terms (3 and -7). Combining the 'x' terms, we have 3x - x, which simplifies to 2x. Combining the constant terms, we have 3 + (-7), which equals -4. Now, we can rewrite the inequality as 2x - 4 > 6. This simplified form is much easier to work with and allows us to focus on isolating 'x'. Simplification is a crucial step because it reduces the complexity of the problem and makes subsequent steps more manageable. It's like decluttering your workspace before starting a task; a clear and organized expression makes the solution process smoother and less prone to errors. This step demonstrates the importance of paying attention to detail and applying basic algebraic principles correctly. By simplifying the inequality, we've set the stage for the next steps, where we'll further isolate 'x' and determine the solution set.

Simplification not only makes the inequality easier to solve but also provides a clearer understanding of the relationship between the variable and the constants. In this case, we've transformed the original inequality into a more concise form, highlighting the key components that influence the solution. The simplified inequality, 2x - 4 > 6, tells us that twice the value of 'x', minus 4, must be greater than 6. This understanding helps us to anticipate the type of solution we're looking for and to verify the reasonableness of our final answer. It's a good practice to always check your simplified expression against the original to ensure that no errors were introduced during the process. This step-by-step approach to simplification is a fundamental skill in algebra and is applicable to a wide range of mathematical problems. By mastering this skill, you'll be well-equipped to tackle more complex inequalities and equations. So, let's move on to the next step, where we'll continue to isolate 'x' and get closer to the solution.

After simplifying the inequality to 2x - 4 > 6, the next step is to isolate the term containing the variable, which in this case is 2x. To do this, we need to eliminate the constant term, -4, from the left side of the inequality. We achieve this by performing the inverse operation, which is adding 4 to both sides of the inequality. Remember, whatever operation we perform on one side of an inequality, we must perform the same operation on the other side to maintain the balance. Adding 4 to both sides of the inequality, we get 2x - 4 + 4 > 6 + 4. This simplifies to 2x > 10. Now, we have successfully isolated the variable term, 2x, on the left side of the inequality. This step is crucial because it brings us closer to solving for 'x' by eliminating the constant term that was interfering with its isolation. The principle of maintaining balance by performing the same operation on both sides is a fundamental concept in solving inequalities and equations alike. It ensures that the relationship between the two sides remains consistent throughout the solution process. By isolating the variable term, we've narrowed our focus to the relationship between 2x and 10, setting the stage for the final step of solving for 'x'.

Isolating the variable term is a critical step in solving inequalities because it allows us to focus solely on the relationship between the variable and the remaining constant term. In our case, 2x > 10 tells us that twice the value of 'x' must be greater than 10. This understanding helps us to anticipate the range of values that 'x' can take and to verify the reasonableness of our final answer. It's important to note that we added 4 to both sides of the inequality, which is the inverse operation of subtraction. If we were dealing with a term being added to the variable term, we would subtract it from both sides. This principle of using inverse operations is a key technique in solving algebraic expressions. By successfully isolating the variable term, we've made significant progress towards solving the inequality. The next and final step will involve isolating 'x' completely by dividing both sides of the inequality by the coefficient of 'x'. So, let's proceed to the final step and determine the solution set for 'x'.

Having isolated the variable term, the final step is to solve for 'x' completely. Our inequality is currently 2x > 10. To isolate 'x', we need to eliminate the coefficient 2, which is multiplying 'x'. We do this by performing the inverse operation, which is dividing both sides of the inequality by 2. Remember, as long as we divide by a positive number, the inequality sign remains the same. If we were dividing by a negative number, we would need to flip the inequality sign. Dividing both sides of the inequality by 2, we get (2x) / 2 > 10 / 2. This simplifies to x > 5. This is our solution! It tells us that 'x' can be any value greater than 5. This range of values satisfies the original inequality, 3x + 3 - x + (-7) > 6. Solving for 'x' is the culmination of the previous steps, where we simplified the inequality, isolated the variable term, and now finally, determined the solution set. The ability to isolate a variable and solve for its value is a fundamental skill in algebra and is essential for solving a wide range of mathematical problems. The solution x > 5 represents an infinite set of values, all of which satisfy the original inequality. This is a key characteristic of inequalities – they often have a range of solutions rather than a single solution.

Now that we've solved for 'x', it's always a good practice to verify our solution. We can do this by choosing a value greater than 5 and substituting it back into the original inequality to see if it holds true. For example, let's choose x = 6. Substituting this value into the original inequality, we get: 3(6) + 3 - 6 + (-7) > 6 18 + 3 - 6 - 7 > 6 8 > 6 This is true, so our solution x > 5 is likely correct. Verification is a crucial step in problem-solving, as it helps to catch any potential errors and ensures the accuracy of our solution. In this case, our verification confirms that our solution is indeed correct. We've successfully navigated the steps of solving the inequality, from simplification to isolation to the final solution. This process demonstrates the importance of a systematic approach to problem-solving, where each step builds upon the previous one to arrive at the correct answer. So, we've confidently solved for 'x' and determined that the solution set is all values greater than 5.

After diligently working through the steps to solve the inequality 3x + 3 - x + (-7) > 6, we have arrived at the solution x > 5. Now, the final task is to identify the correct option from the given choices: A) x < 5 B) x > 5 C) x > (-5) D) x > 2.5. Comparing our solution, x > 5, with the options, it is clear that option B, x > 5, matches our result perfectly. Therefore, option B is the correct answer. This step highlights the importance of careful comparison and attention to detail. We've spent time and effort solving the inequality, and it's crucial to ensure that we select the correct answer from the given options. A simple mistake in selecting the option can negate all the hard work we've put in. The process of identifying the correct option also reinforces our understanding of the solution. By comparing our solution with the options, we solidify our grasp of what the inequality represents and the range of values that satisfy it. This is a valuable step in the learning process, as it helps us to internalize the concepts and apply them to future problems.

The ability to identify the correct answer from a set of options is a crucial skill in mathematics and in many other areas of life. It requires not only a thorough understanding of the problem and its solution but also the ability to carefully analyze and compare different options. In this case, we were presented with four options, each representing a different range of values for 'x'. By systematically comparing our solution with each option, we were able to confidently identify the correct answer. This process demonstrates the importance of a methodical approach to problem-solving, where each step is executed with precision and attention to detail. We started by simplifying the inequality, then isolated the variable term, solved for 'x', and finally, identified the correct option. This step-by-step approach not only ensures accuracy but also helps us to develop a deeper understanding of the underlying concepts. So, with confidence, we can conclude that option B, x > 5, is the correct solution to the inequality 3x + 3 - x + (-7) > 6.

In conclusion, we have successfully navigated the process of solving the inequality 3x + 3 - x + (-7) > 6, arriving at the solution x > 5 and correctly identifying option B as the answer. This journey through the steps of simplification, isolation, and solution highlights the importance of a systematic approach to mathematical problem-solving. Each step builds upon the previous one, leading us closer to the final answer. The ability to solve inequalities is a fundamental skill in mathematics, with applications in various fields, from engineering to economics. Inequalities allow us to define ranges and conditions for solutions, providing a more nuanced understanding of mathematical relationships. Mastering inequalities requires a solid grasp of algebraic principles, including the rules for manipulating expressions and maintaining balance. It also requires careful attention to detail and the ability to apply the correct operations in the correct order. The key takeaway from this exercise is that solving inequalities, like solving equations, is a process that can be broken down into manageable steps. By following a systematic approach and paying attention to detail, we can confidently tackle even complex inequalities and arrive at the correct solution.

The skills we've developed in solving this inequality are transferable to a wide range of mathematical problems. The principles of simplification, isolation, and verification are applicable to various algebraic expressions and equations. Furthermore, the ability to identify the correct answer from a set of options is a valuable skill in any problem-solving context. Inequalities are not just abstract mathematical concepts; they have real-world applications in various fields. For example, in engineering, inequalities are used to define tolerances and safety margins. In economics, they are used to model supply and demand relationships. In computer science, they are used to analyze algorithms and data structures. By mastering inequalities, we equip ourselves with a powerful tool for understanding and solving problems in a variety of contexts. This article has provided a comprehensive guide to solving a specific inequality, but the principles and techniques discussed here can be applied to many other inequalities. With practice and perseverance, we can all master the art of solving inequalities and unlock their potential for problem-solving and decision-making.