Solving For X In The Equation \frac{1}{2}-x+\frac{3}{2}=x-4

Step 1: Combine Like Terms on the Left Side

The initial step in simplifying the equation $\frac{1}{2}-x+\frac{3}{2}=x-4$ involves combining the like terms on the left side of the equation. Like terms are terms that have the same variable raised to the same power, or constants. In this case, we have two constant terms, $\frac{1}{2}$ and $\frac{3}{2}$, which can be combined. Adding these fractions is straightforward because they share a common denominator. $\frac{1}{2} + \frac{3}{2}$ equals $\frac{4}{2}$, which simplifies to 2. Therefore, the left side of the equation now becomes $2 - x$. This simplification is a crucial step as it reduces the complexity of the equation, making it easier to manipulate and solve. By combining the constants, we've effectively reduced the number of terms on the left side, paving the way for isolating the variable x. This process highlights the importance of recognizing and combining like terms in algebraic expressions, a skill that is fundamental for solving a wide range of mathematical problems. Remember, the goal is always to simplify the equation as much as possible before proceeding to further steps. This not only makes the calculations easier but also reduces the chances of making errors.

Step 2: Add x to Both Sides of the Equation

Having simplified the left side of the equation to $2 - x$, our next objective is to isolate the variable x. To achieve this, we need to eliminate the -x term from the left side. The golden rule of equation solving is that whatever operation you perform on one side, you must perform the same operation on the other side to maintain the equality. In this case, we will add x to both sides of the equation. Adding x to the left side, $2 - x + x$, cancels out the -x term, leaving us with just 2. On the right side, we have x - 4, and adding x gives us x - 4 + x, which simplifies to 2x - 4. So, the equation now transforms to 2 = 2x - 4. This step is pivotal because it moves all the x terms to one side of the equation, bringing us closer to isolating x. By adding x to both sides, we've effectively balanced the equation while simultaneously simplifying it. This technique is a cornerstone of algebraic manipulation and is essential for solving equations efficiently. Understanding this principle allows you to confidently manipulate equations and move terms around without altering the solution. Remember, the key is to maintain balance and perform the same operation on both sides.

Step 3: Add 4 to Both Sides of the Equation

Following the previous step, our equation now stands as 2 = 2x - 4. To further isolate x, we need to eliminate the constant term -4 from the right side. Again, we apply the principle of performing the same operation on both sides to maintain equality. This time, we add 4 to both sides of the equation. On the left side, 2 + 4 equals 6. On the right side, 2x - 4 + 4 simplifies to 2x, as the -4 and +4 cancel each other out. This transforms our equation to 6 = 2x. This step is crucial as it isolates the term containing x on one side of the equation, setting us up for the final step of solving for x. By adding 4 to both sides, we've effectively neutralized the constant term on the right, bringing us closer to our goal. This process underscores the importance of strategic manipulation in equation solving. Each step is carefully chosen to simplify the equation and isolate the variable of interest. Remember, the aim is to systematically peel away the layers surrounding x until it stands alone, revealing its value.

Step 4: Divide Both Sides by 2

Our equation has now been simplified to 6 = 2x. The final step in solving for x is to isolate x completely. Currently, x is being multiplied by 2. To undo this multiplication, we perform the inverse operation, which is division. We divide both sides of the equation by 2. On the left side, 6 divided by 2 equals 3. On the right side, 2x divided by 2 simplifies to x. Therefore, our equation becomes 3 = x. This step is the climax of our solving process, as it reveals the value of x. By dividing both sides by 2, we've effectively isolated x, giving us the solution to the equation. This demonstrates the power of inverse operations in algebra. Each operation has an inverse that undoes its effect, allowing us to systematically isolate variables and solve equations. Remember, the goal is to get x by itself on one side of the equation, and division is the key to unlocking x in this case. This final step confirms that the solution to the equation $\frac{1}{2}-x+\frac{3}{2}=x-4$ is indeed x = 3.

Conclusion

In conclusion, solving the equation $\frac{1}{2}-x+\frac{3}{2}=x-4$ involves a series of strategic steps aimed at isolating the variable x. We began by combining like terms, then strategically added and subtracted terms from both sides to simplify the equation. Finally, we used division to completely isolate x and arrive at the solution. The correct answer is B. x = 3. This process highlights the importance of understanding algebraic principles and applying them systematically. Each step is crucial, and mastering these techniques is essential for success in mathematics. Equation solving is a fundamental skill that extends far beyond the classroom, finding applications in various fields such as science, engineering, and finance. By understanding the underlying logic and practicing these techniques, you can confidently tackle more complex mathematical problems. Remember, the key is to break down the problem into smaller, manageable steps and apply the appropriate operations to maintain balance and isolate the variable of interest. With practice and patience, you can become proficient in solving equations and unlock the power of algebra.