Solving For X In -x + 3/2 = X - 4 A Step-by-Step Guide
In the realm of mathematics, solving equations is a fundamental skill. This article delves into the process of finding the value of 'x' in the equation -x + 3/2 = x - 4. We will explore the steps involved, provide a clear explanation, and arrive at the correct solution. This exploration will not only solidify your understanding of algebraic equations but also enhance your problem-solving abilities.
Understanding the Problem
At the heart of our discussion lies the equation: -x + 3/2 = x - 4. To solve for 'x', our primary goal is to isolate 'x' on one side of the equation. This involves a series of algebraic manipulations that maintain the equation's balance. Remember, whatever operation we perform on one side of the equation, we must perform the same operation on the other side to preserve equality. Before we dive into the solution, let's break down the equation's components. We have 'x' terms on both sides, a constant term (3/2) on the left, and another constant term (-4) on the right. Our strategy will involve combining like terms to simplify the equation and eventually isolate 'x'. This process requires a solid grasp of basic algebraic principles, including the addition and subtraction properties of equality, as well as the concept of inverse operations. By understanding these underlying principles, we can confidently navigate the steps to solve for 'x'.
Step-by-Step Solution
Our first step in solving for x in the equation -x + 3/2 = x - 4 is to gather all the 'x' terms on one side of the equation. To achieve this, we can add 'x' to both sides. This eliminates the '-x' term on the left side, resulting in the equation 3/2 = 2x - 4. This manipulation is based on the addition property of equality, which states that adding the same value to both sides of an equation preserves the equality. Next, we need to isolate the term with 'x'. We can do this by adding 4 to both sides of the equation. This eliminates the '-4' on the right side, giving us 3/2 + 4 = 2x. Simplifying the left side, we get 11/2 = 2x. Now, to finally isolate 'x', we divide both sides of the equation by 2. This gives us x = (11/2) / 2, which simplifies to x = 11/4. Therefore, the solution to the equation -x + 3/2 = x - 4 is x = 11/4. This step-by-step approach ensures that we maintain the balance of the equation throughout the process, leading us to the correct solution. Remember, each step is based on fundamental algebraic principles, and understanding these principles is crucial for solving more complex equations.
Detailed Breakdown of Each Step
Let's delve deeper into each step of solving the equation -x + 3/2 = x - 4, ensuring a comprehensive understanding. The initial equation presents 'x' terms on both sides, hindering our ability to directly determine its value. Our first task is to consolidate these 'x' terms. We achieve this by adding 'x' to both sides of the equation. This action eliminates '-x' from the left side, leaving us with 3/2. On the right side, 'x' combines with the added 'x' to become '2x'. The equation now transforms to 3/2 = 2x - 4. This step leverages the addition property of equality, a cornerstone of algebraic manipulation. Next, we focus on isolating the term containing 'x'. The constant term '-4' on the right side obstructs this isolation. To counter this, we add '4' to both sides of the equation. This cancels out the '-4' on the right, while on the left, we have the expression 3/2 + 4. This addition results in 11/2, and our equation progresses to 11/2 = 2x. Now, 'x' is only multiplied by '2'. To completely isolate 'x', we perform the inverse operation – division. Dividing both sides of the equation by '2' effectively undoes the multiplication. On the right, '2x' divided by '2' simplifies to 'x'. On the left, 11/2 divided by '2' yields 11/4. Thus, we arrive at the solution: x = 11/4. Each step in this process is deliberate, grounded in algebraic principles, and aimed at systematically isolating 'x'.
Identifying the Correct Answer
After meticulously solving the equation -x + 3/2 = x - 4, we arrived at the solution x = 11/4. Now, let's examine the provided options to determine the correct answer. The options presented are:
A. x = 3 B. x = 1/3 C. x = -1/3 D. x = -3
By comparing our solution, x = 11/4, with the given options, it becomes evident that none of the options directly match our result. However, it's crucial to recognize that 11/4 is a fraction, while the options are presented as integers or fractions with a denominator of 3. To accurately compare, we can convert 11/4 into a mixed number or a decimal. Converting 11/4 to a mixed number gives us 2 3/4, and converting it to a decimal gives us 2.75. Clearly, none of the provided options (3, 1/3, -1/3, -3) are equivalent to 11/4, 2 3/4, or 2.75. Therefore, based on the given options, the correct answer is none of the above. It's essential to be thorough in comparing the solution with the options, ensuring that we consider different representations of the same value. This careful comparison highlights the importance of accuracy in both solving the equation and identifying the correct answer from the given choices.
Common Mistakes to Avoid
When solving equations like -x + 3/2 = x - 4, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can significantly improve your problem-solving accuracy. One frequent error is failing to apply operations to both sides of the equation. Remember, the principle of equality dictates that any operation performed on one side must be mirrored on the other. For instance, if you add 'x' to the left side, you must also add 'x' to the right side. Neglecting this balance will disrupt the equation and yield a wrong solution. Another common mistake arises when dealing with fractions. Students sometimes struggle with adding or subtracting fractions, particularly when they have different denominators. In our equation, we encountered the term 3/2. Incorrectly handling this fraction during the simplification process can lead to an inaccurate result. It's crucial to ensure that fractions are added or subtracted correctly by finding a common denominator. Sign errors also frequently occur, especially when dealing with negative terms. For example, students might incorrectly add or subtract negative numbers, leading to an incorrect sign in the final answer. Careful attention to the rules of sign manipulation is essential. Finally, a mistake can occur during the simplification process. Overlooking a step or incorrectly combining like terms can derail the solution. It's always a good practice to double-check each step to ensure accuracy. By being mindful of these common mistakes and taking the time to review your work, you can minimize errors and confidently arrive at the correct solution.
Alternative Approaches to Solving the Equation
While we've explored a step-by-step method to solve the equation -x + 3/2 = x - 4, it's beneficial to consider alternative approaches. These different perspectives can enhance your problem-solving flexibility and deepen your understanding of algebraic manipulations. One alternative approach involves rearranging the equation in a slightly different order. Instead of initially adding 'x' to both sides, we could subtract 'x' from both sides. This would result in the equation -2x + 3/2 = -4. From here, we could subtract 3/2 from both sides, leading to -2x = -4 - 3/2, which simplifies to -2x = -11/2. Finally, dividing both sides by -2 would give us x = 11/4, the same solution we obtained previously. This demonstrates that the order in which we perform operations can vary without affecting the final result. Another approach involves eliminating the fraction early in the process. To do this, we could multiply both sides of the original equation by the denominator, 2. This would give us -2x + 3 = 2x - 8. Now, we have an equation without fractions, which some might find easier to manipulate. We could then add 2x to both sides, resulting in 3 = 4x - 8. Adding 8 to both sides gives us 11 = 4x, and finally, dividing by 4 yields x = 11/4. These alternative approaches highlight the versatility of algebraic techniques. By understanding different methods, you can choose the one that best suits your individual problem-solving style and the specific characteristics of the equation.
Conclusion
In conclusion, we have thoroughly explored the solution to the equation -x + 3/2 = x - 4. Through a step-by-step approach, we identified the correct solution as x = 11/4. We also examined common mistakes to avoid and explored alternative solution methods. A strong foundation in algebra is crucial for tackling more complex mathematical problems. The ability to manipulate equations, isolate variables, and solve for unknowns is a skill that extends far beyond the classroom. It is applicable in various fields, including science, engineering, economics, and computer science. By mastering these fundamental concepts, you equip yourself with valuable tools for problem-solving in diverse contexts. Remember, practice is key to solidifying your understanding. The more you engage with algebraic equations, the more confident and proficient you will become in solving them. So, continue to challenge yourself with new problems, explore different approaches, and deepen your mathematical knowledge. The journey of learning mathematics is a continuous one, and each equation solved brings you one step closer to mastery.
