Solving For X Step-by-Step The Equation (1/2) - X + (3/2) = X - 4

In the realm of mathematics, solving algebraic equations is a fundamental skill that serves as a cornerstone for more advanced concepts. Equations, in their essence, represent relationships between variables and constants, and the ability to manipulate and solve them is crucial in various fields, from science and engineering to economics and finance. This article delves into the process of solving a specific linear equation, providing a step-by-step guide to isolate the unknown variable, x. Mastering the techniques involved not only enhances problem-solving abilities but also lays a solid foundation for tackling more complex mathematical challenges. So, let's embark on this journey of algebraic exploration and unravel the mystery behind the equation $\frac{1}{2}-x+\frac{3}{2}=x-4$. This introduction sets the stage for understanding the significance of equation solving and prepares the reader for a detailed walkthrough of the solution process. Understanding how to solve for unknowns in equations is a critical skill in mathematics. Today, we will be working through the equation $\frac{1}{2}-x+\frac{3}{2}=x-4$ to find the correct value for $x$. We will explore the step-by-step process to arrive at the accurate solution and understand the underlying algebraic principles. This article aims to clarify each step, making it easy to follow and comprehend the solution. Solving algebraic equations is a fundamental skill, and mastering it will significantly benefit your mathematical journey. The ability to manipulate equations and isolate variables is key to success in various mathematical disciplines. This article focuses on a specific equation, $\frac{1}{2}-x+\frac{3}{2}=x-4$, providing a detailed walkthrough of the solution process. By understanding each step and the underlying principles, you will enhance your problem-solving skills and gain confidence in tackling similar challenges. This comprehensive guide aims to demystify the process and empower you with the tools to solve equations effectively. Whether you're a student learning algebra or someone looking to refresh your mathematical skills, this article will provide valuable insights and a clear understanding of how to solve for x. The importance of understanding how to solve for x in equations cannot be overstated. This fundamental skill is the bedrock of algebra and is crucial for progressing in mathematics and related fields. Algebraic equations are mathematical statements that assert the equality of two expressions, and solving them involves finding the value(s) of the unknown variable(s) that make the equation true. In this article, we will dissect the equation $\frac{1}{2}-x+\frac{3}{2}=x-4$, providing a detailed, step-by-step solution to determine the value of x. We will not only present the solution but also explain the underlying algebraic principles, ensuring that you grasp the logic behind each step. This approach will equip you with the tools to solve a wide range of equations, fostering a deeper understanding of mathematical concepts. Whether you are a student, educator, or simply someone with a passion for mathematics, this article will serve as a valuable resource for mastering the art of solving algebraic equations.

Step 1: Combining Like Terms on the Left Side

To begin solving the equation $\frac{1}{2}-x+\frac{3}{2}=x-4$, our first step is to simplify the left side by combining like terms. This involves adding the constant terms $\frac{1}{2}$ and $\frac{3}{2}$. The process of combining like terms is a cornerstone of algebraic simplification, allowing us to condense expressions and make them easier to manipulate. In this case, we are dealing with fractions, so it's essential to have a solid understanding of fraction addition. Adding fractions with a common denominator is straightforward – we simply add the numerators and keep the denominator the same. This principle allows us to combine the constant terms efficiently and move closer to isolating the variable x. By simplifying one side of the equation, we reduce the complexity of the problem and pave the way for further steps in the solution process. This initial step is crucial for organizing the equation and setting the stage for subsequent algebraic manipulations. Combining like terms is a fundamental algebraic technique that simplifies equations by grouping similar terms together. In the given equation, $\frac{1}{2}-x+\frac{3}{2}=x-4$, we identify $\frac{1}{2}$ and $\frac{3}{2}$ as like terms because they are both constants. To combine them, we perform the addition $\frac{1}{2} + \frac{3}{2}$. Since they have a common denominator, the process is straightforward: we add the numerators (1 + 3) and keep the denominator (2). This results in $\frac{4}{2}$, which simplifies to 2. Thus, the left side of the equation becomes $2 - x$. This simplification is a critical first step as it reduces the equation's complexity, making it easier to isolate the variable x in subsequent steps. Mastering the combination of like terms is essential for solving various algebraic equations. This technique is not only applicable to simple equations but also to more complex expressions involving polynomials and rational functions. By consistently applying this method, you can effectively streamline your problem-solving process and avoid common errors. In our specific equation, combining the constants on the left side helps us consolidate the numerical terms, allowing us to focus on the variable x. This strategic approach to simplification is key to successfully navigating algebraic challenges. Remember, a well-organized equation is easier to solve, and combining like terms is a primary method for achieving that organization. The principle of combining like terms is a cornerstone of algebraic manipulation, and its effective application is crucial for solving equations efficiently. When faced with an equation such as $\frac{1}{2}-x+\frac{3}{2}=x-4$, the first step should always be to identify and combine any terms that share the same variable or are constants. In this case, the constant terms $\frac{1}{2}$ and $\frac{3}{2}$ can be combined. To do this, we simply add the two fractions together. Since they already have a common denominator of 2, the addition is straightforward: $\frac{1}{2} + \frac{3}{2} = \frac{1+3}{2} = \frac{4}{2}$. This fraction can be further simplified to 2. Thus, the left side of the equation, $\frac{1}{2}-x+\frac{3}{2}$, simplifies to $2 - x$. This step is essential because it reduces the number of terms in the equation, making it more manageable. By combining like terms, we create a more streamlined expression that is easier to work with in subsequent steps. This simplification is a key strategy in solving algebraic equations and is applicable to a wide range of problems. Mastering this technique will greatly enhance your ability to solve for unknowns and navigate complex mathematical expressions with greater confidence and accuracy.

Step 2: Rewrite the Equation

After combining the constants on the left side, the equation now looks like $2 - x = x - 4$. This rewritten form is crucial because it presents a clearer picture of the equation's structure. By simplifying the left side, we have reduced the complexity and made it easier to visualize the next steps required to isolate x. Rewriting the equation is not just a matter of tidying up; it's about transforming it into a more workable state. This step highlights the importance of algebraic manipulation in problem-solving. A well-organized equation is significantly easier to solve than a cluttered one. The rewritten equation $2 - x = x - 4$ sets the stage for the subsequent steps, which will involve moving terms across the equals sign to gather the x terms on one side and the constants on the other. This strategic approach is fundamental to solving linear equations. The goal of rewriting the equation is to create a clearer, more manageable form that facilitates further steps in the solution process. In this case, simplifying the left side by combining the constants resulted in the equation $2 - x = x - 4$. This form allows us to see the relationship between the variable x and the constants more clearly. It also sets the stage for the next crucial steps: isolating the x terms on one side and the constant terms on the other. Rewriting equations is not just a matter of aesthetics; it's a strategic move that enhances our ability to solve them. A well-structured equation is easier to manipulate and understand, reducing the chances of errors and promoting a smoother problem-solving experience. By taking the time to rewrite and simplify, we are investing in a more efficient and accurate solution. The equation $2 - x = x - 4$ now presents a balanced view, with the variable appearing on both sides and the constants clearly visible. This balanced representation is a stepping stone to isolating x and finding its value. The act of rewriting an equation after performing initial simplifications is a critical step in the problem-solving process. It’s not merely about making the equation look neater; it’s about strategically positioning it for the next phase of manipulation. In the case of our equation, $\frac{1}{2}-x+\frac{3}{2}=x-4$, after combining the constants on the left side, we arrived at the simplified form $2 - x = x - 4$. This rewritten equation is significantly clearer and more manageable than the original. It allows us to visualize the next steps required to isolate the variable x and solve for its value. The act of rewriting often involves rearranging terms, combining like terms, or applying other algebraic operations to transform the equation into a more convenient form. This process is akin to laying the groundwork for a construction project; a solid foundation is essential for a successful outcome. In this rewritten form, the terms are more clearly separated, and the relationship between the variable and the constants is more apparent. This clarity is crucial for making informed decisions about the subsequent steps. For example, we can now easily see that we need to move the x terms to one side of the equation and the constants to the other. This strategic perspective is only possible because we took the time to rewrite the equation. The rewritten equation, $2 - x = x - 4$, is not just a cleaner version of the original; it’s a stepping stone to the solution. It’s a testament to the power of strategic algebraic manipulation and the importance of taking a thoughtful approach to problem-solving.

Step 3: Add x to Both Sides

To further isolate x, the next step is to add x to both sides of the equation $2 - x = x - 4$. This is a crucial step because it eliminates the x term on the left side, effectively moving all x terms to the right side of the equation. The principle behind adding the same quantity to both sides of an equation is rooted in the fundamental property of equality. This property ensures that the balance of the equation is maintained, allowing us to perform algebraic manipulations without altering the solution. By adding x to both sides, we are essentially undoing the subtraction of x on the left side, bringing us closer to isolating the variable. This step demonstrates the power of inverse operations in solving equations. Adding x is the inverse operation of subtracting x, and by applying this principle, we simplify the equation and make it more amenable to further manipulation. The resulting equation will have all the x terms on one side, setting the stage for isolating x and finding its value. Adding x to both sides of the equation $2 - x = x - 4$ is a pivotal step in isolating the variable. This action leverages the addition property of equality, which states that adding the same quantity to both sides of an equation maintains the equality. By adding x to both sides, we aim to eliminate the $-x$ term on the left side and consolidate the x terms on the right side. When we add x to the left side, $2 - x + x$, the $-x$ and $+x$ terms cancel each other out, leaving us with just 2. On the right side, we have $x - 4 + x$, which simplifies to $2x - 4$. Therefore, the equation now becomes $2 = 2x - 4$. This transformation is significant because it gathers all the x terms on one side, bringing us closer to solving for x. This step exemplifies a common strategy in algebra: using inverse operations to isolate variables. The inverse operation of subtraction is addition, and by adding x to both sides, we effectively reversed the subtraction of x on the left side. This technique is fundamental to solving various algebraic equations and demonstrates the importance of understanding the properties of equality. The resulting equation, $2 = 2x - 4$, is now in a form that is easier to manipulate. We have successfully moved all the x terms to the right side, and the next steps will involve isolating x by addressing the constant term on the right side. The act of adding x to both sides of the equation $2 - x = x - 4$ is a strategic move rooted in the principles of algebraic manipulation. This step serves a crucial purpose: to consolidate the terms involving the variable x onto one side of the equation. By doing so, we pave the way for isolating x and ultimately finding its value. The underlying principle at play here is the addition property of equality. This property states that if you add the same quantity to both sides of an equation, the equality remains valid. In our case, we are adding x to both sides. On the left side, we have $2 - x$. Adding x to this gives us $2 - x + x$. The $-x$ and $+x$ terms cancel each other out, leaving us with just 2. On the right side, we have $x - 4$. Adding x to this gives us $x - 4 + x$. Combining the x terms, we get $2x - 4$. So, after adding x to both sides, our equation transforms into $2 = 2x - 4$. This transformation is significant because it brings all the x terms to the right side of the equation. Now, we have a clearer picture of the relationship between x and the constants. The next steps will involve isolating x by addressing the constant term on the right side. This strategic manipulation of the equation is a hallmark of algebraic problem-solving. By understanding and applying the properties of equality, we can effectively rearrange equations to isolate variables and find their values. The step of adding x to both sides is a perfect illustration of this principle in action.

Step 4: Add 4 to Both Sides

Following the addition of x to both sides, our equation now stands as $2 = 2x - 4$. The next logical step in isolating x is to address the constant term on the right side, which is -4. To do this, we apply the inverse operation of subtraction, which is addition. By adding 4 to both sides of the equation, we effectively eliminate the -4 on the right side, moving us closer to isolating the x term. This step further exemplifies the use of the addition property of equality. Just as adding x to both sides maintained the balance of the equation, adding 4 to both sides also preserves the equality. This consistent application of algebraic principles is crucial for solving equations accurately. Adding 4 to both sides simplifies the equation by isolating the term containing x on the right side. The resulting equation will have a constant term on the left side and a multiple of x on the right side, setting the stage for the final step of dividing to solve for x. The strategic manipulation of equations through inverse operations is a cornerstone of algebra. Adding 4 to both sides of the equation $2 = 2x - 4$ is a critical step in isolating the x term. This action utilizes the principle of inverse operations to counteract the subtraction of 4 on the right side. By adding 4 to both sides, we are effectively moving the constant term from the right side to the left side, bringing us closer to solving for x. The process of adding 4 to both sides is guided by the addition property of equality. This fundamental property ensures that if we add the same value to both sides of an equation, the equality remains valid. This allows us to manipulate equations without changing their solutions. When we add 4 to the left side of the equation, $2 + 4$, we get 6. On the right side, we have $2x - 4 + 4$. The -4 and +4 terms cancel each other out, leaving us with just $2x$. Therefore, the equation now becomes $6 = 2x$. This transformation is significant because it has isolated the x term on the right side, with only a coefficient attached to it. The next step will involve dividing both sides by this coefficient to solve for x. The act of adding 4 to both sides highlights the power of inverse operations in algebra. Subtraction and addition are inverse operations, and by using addition to counteract subtraction, we have successfully simplified the equation. This technique is a common strategy in solving various algebraic problems and demonstrates the importance of understanding the relationships between mathematical operations. The resulting equation, $6 = 2x$, is now in a simplified form that is easily solvable. We have successfully isolated the x term and are just one step away from finding its value. This progress is a testament to the effectiveness of applying algebraic principles strategically and consistently. The step of adding 4 to both sides of the equation $2 = 2x - 4$ is a pivotal moment in the process of solving for x. This seemingly simple action is a powerful application of the addition property of equality, a cornerstone of algebraic manipulation. The goal here is clear: to isolate the term containing x on one side of the equation, and by adding 4 to both sides, we take a significant step toward achieving this. The rationale behind adding 4 is to counteract the subtraction of 4 that is already present on the right side of the equation. Remember, mathematical operations often come in pairs – addition and subtraction, multiplication and division – and using one to undo the other is a fundamental strategy in solving equations. When we add 4 to the left side, $2 + 4$, we arrive at 6. This is a straightforward arithmetic operation. On the right side, we have $2x - 4 + 4$. Here, the -4 and +4 terms are additive inverses; they cancel each other out, leaving us with just $2x$. This cancellation is precisely what we aimed for. The equation now stands as $6 = 2x$. This new form is significantly closer to our goal of isolating x. We have successfully eliminated the constant term on the right side, leaving only the term with x. The next and final step will be to divide both sides by the coefficient of x. The strategic use of the addition property of equality has brought us to this point. By understanding and applying this property, we can confidently manipulate equations, knowing that we are maintaining the balance and not altering the solution. The step of adding 4 to both sides is a clear demonstration of the power and elegance of algebraic techniques.

Step 5: Divide Both Sides by 2

With the equation simplified to $6 = 2x$, the final step in solving for x is to divide both sides by 2. This action isolates x on the right side, revealing its value. The principle behind dividing both sides of an equation by the same non-zero number is another application of the fundamental property of equality. This property ensures that the balance of the equation is maintained throughout the algebraic manipulation. Dividing by 2 is the inverse operation of multiplying by 2, which is the operation currently applied to x on the right side. By applying this inverse operation, we undo the multiplication and isolate x. This step showcases the importance of understanding inverse operations in solving equations. The result of dividing both sides by 2 is a clear and concise solution for x. This final step brings the problem to a satisfying conclusion, demonstrating the effectiveness of the step-by-step algebraic process we have followed. Dividing both sides of the equation $6 = 2x$ by 2 is the concluding step in isolating x and finding its value. This action is based on the division property of equality, which asserts that dividing both sides of an equation by the same non-zero number preserves the equality. The purpose of this division is to undo the multiplication of x by 2, effectively isolating x on one side of the equation. When we divide the left side, 6, by 2, we get 3. On the right side, we have $2x$ divided by 2. The 2 in the numerator and the 2 in the denominator cancel each other out, leaving us with just x. Therefore, the equation becomes $3 = x$, or equivalently, $x = 3$. This result is the solution to the original equation. It represents the value of x that makes the equation true. The act of dividing both sides by 2 demonstrates the power of inverse operations in solving equations. Multiplication and division are inverse operations, and by using division to counteract multiplication, we have successfully isolated the variable. This technique is a cornerstone of algebraic problem-solving and is applicable to a wide range of equations. The final equation, $x = 3$, provides a clear and concise answer. We have successfully navigated the steps required to solve for x and have arrived at a definitive solution. This process highlights the importance of understanding algebraic principles and applying them strategically to solve mathematical problems. The concluding step of dividing both sides of the equation $6 = 2x$ by 2 is the final flourish in our algebraic dance, the move that reveals the sought-after value of x. This action is a direct application of the division property of equality, a principle that, like its addition counterpart, ensures that our manipulation maintains the equation's balance. The goal here is crystal clear: to isolate x completely, to strip away the coefficient that clings to it and reveal its solitary value. The coefficient in this case is 2, and the operation connecting it to x is multiplication. To undo multiplication, we employ its inverse: division. When we divide the left side, 6, by 2, the arithmetic is straightforward: $6 / 2 = 3$. On the right side, we have $2x$ divided by 2. This is where the magic of inverse operations truly shines. The 2 in the numerator cancels out the 2 in the denominator, leaving us with just x. The equation now proudly proclaims: $3 = x$. We have arrived at our destination. x stands alone, its value unveiled. We can confidently rewrite this as $x = 3$, a solution that satisfies the original equation. This final division is not just a mechanical step; it's a testament to the power of algebraic reasoning. By understanding and applying the division property of equality, we have navigated the equation with precision, each step building upon the last, until we reached our objective. The solution, $x = 3$, is the culmination of our efforts, a clear and concise answer that validates our journey through the world of algebra.

Conclusion: The Solution and its Significance

Therefore, the solution to the equation $\frac{1}{2}-x+\frac{3}{2}=x-4$ is $x = 3$, which corresponds to option B. This solution represents the value of x that satisfies the equation, making the left side equal to the right side. The process of solving this equation highlights several key algebraic principles, including combining like terms, applying the addition property of equality, and using inverse operations to isolate the variable. Mastering these principles is crucial for success in algebra and other areas of mathematics. The ability to solve equations is not just an academic exercise; it has practical applications in various fields, from science and engineering to economics and finance. Understanding how to manipulate equations and find solutions is a valuable skill that can be applied to real-world problems. The solution we have found, $x = 3$, is not just a number; it's a testament to the power of algebraic reasoning and the effectiveness of systematic problem-solving. Therefore, the final answer to the equation $\frac{1}{2}-x+\frac{3}{2}=x-4$ is B. $x=3$. This result is the culmination of a step-by-step process, where each action was carefully considered to isolate x and determine its value. The significance of this solution lies not only in the answer itself but also in the method used to obtain it. By following algebraic principles such as combining like terms, adding the same quantity to both sides, and using inverse operations, we have demonstrated a robust approach to problem-solving. This method can be applied to a wide range of equations, making it a valuable tool in mathematics and related disciplines. The value $x = 3$ is the unique solution to the equation, meaning that it is the only value that will make the equation true. This highlights the precision of mathematical solutions and the importance of accurate calculations. In conclusion, the solution $x = 3$ is a testament to the power of algebra and the importance of mastering its principles. It represents a successful application of problem-solving techniques and a clear understanding of how to manipulate equations. Therefore, the solution $x = 3$ to the equation $\frac{1}{2}-x+\frac{3}{2}=x-4$ is more than just a numerical answer; it's a demonstration of the power of algebra and the significance of a systematic approach to problem-solving. Throughout this article, we have meticulously walked through each step, highlighting the underlying principles and the rationale behind each action. From combining like terms to applying the addition and division properties of equality, we have showcased the tools that make algebraic manipulation possible. The final answer, $x = 3$, is the tangible result of this process. It is the value that, when substituted into the original equation, makes the equation true. This verification is a crucial step in any problem-solving endeavor, ensuring that our solution is accurate and reliable. But the significance of the solution extends beyond its numerical value. It lies in the understanding and application of algebraic principles. By mastering these principles, we gain the ability to solve not just this specific equation, but a wide range of mathematical problems. The systematic approach we have employed is a transferable skill, applicable to various disciplines and real-world scenarios. The ability to break down a complex problem into smaller, manageable steps, to identify and apply the relevant principles, and to arrive at a logical solution is a valuable asset in any field. In conclusion, the solution $x = 3$ is a testament to the power of algebra and the importance of developing strong problem-solving skills. It is a reminder that mathematics is not just about numbers and symbols; it's about logical reasoning, critical thinking, and the ability to navigate challenges with confidence and precision.