Solving (x^2 - 3x + 2) / (x - 1) = X - 2 Algebraically A Comprehensive Guide

Introduction

The equation ${ \frac{x^2 - 3x + 2}{x - 1} = x - 2 }$ presents an interesting problem that can be approached in multiple ways. Algebraically, we can manipulate the equation to find the solution set. Alternatively, we can view it graphically as the intersection of two functions, providing a visual understanding of the solution. In this article, we will primarily focus on solving the equation algebraically, detailing each step to ensure clarity. Furthermore, we will touch upon the graphical interpretation to provide a comprehensive understanding of the problem. This exploration is crucial for anyone studying algebra, as it reinforces the importance of considering multiple approaches to problem-solving. Understanding the algebraic solution not only provides the answer but also enhances your skills in manipulating equations, factoring, and identifying potential pitfalls such as extraneous solutions. By mastering such techniques, you will be better equipped to tackle more complex mathematical problems. This article aims to provide a thorough guide on solving the equation, ensuring that you grasp every concept involved. Moreover, the discussion will highlight the significance of checking for extraneous solutions, which are common when dealing with rational equations. This is an essential step in ensuring the accuracy of your solution. By the end of this discussion, you will be able to confidently solve similar equations and understand the nuances involved in the process. Solving equations like this involves a combination of algebraic manipulation and careful consideration of domain restrictions. It’s not just about finding an answer; it’s about understanding the underlying principles and how they apply. So, let’s dive in and explore the fascinating world of algebraic equations and their solutions.

Algebraic Solution

To solve the equation ${ \frac{x^2 - 3x + 2}{x - 1} = x - 2 }$ algebraically, we'll follow a step-by-step approach. The initial step involves simplifying the rational expression by factoring the numerator. Factoring the quadratic expression ${ x^2 - 3x + 2 }$ is a critical part of this process. The ability to factor quadratics accurately is a fundamental skill in algebra, and it's crucial for simplifying complex equations. In this case, we look for two numbers that multiply to 2 and add to -3. These numbers are -1 and -2. Thus, the quadratic expression can be factored as ${ (x - 1)(x - 2) }$. Understanding how to factor quadratic expressions is essential for simplifying rational expressions and solving equations. It is a skill that is frequently used in various areas of mathematics. Once the numerator is factored, the equation becomes ${ \frac{(x - 1)(x - 2)}{x - 1} = x - 2 }$. This step sets the stage for further simplification by canceling common factors. However, it’s important to remember that we must consider the values of x that would make the denominator zero. In this case, x cannot be 1, as this would lead to division by zero, which is undefined. This restriction is crucial and must be taken into account when determining the final solution. The next step involves canceling the common factor of ${ (x - 1) }$ in the numerator and denominator. This simplification is valid as long as ${ x \neq 1 }$. Canceling common factors is a common technique in algebra, but it's essential to be mindful of the restrictions it introduces. After canceling the factor, the equation simplifies to ${ x - 2 = x - 2 }$. This simplified equation highlights an important aspect of the problem, which we will discuss in detail later.

Step-by-Step Solution

1. Factor the numerator:

   The numerator ${ x^2 - 3x + 2 }$ can be factored into ${ (x - 1)(x - 2) }$. This factorization is a key step in simplifying the rational expression. Accurate factorization is crucial for solving algebraic equations. When we factor the numerator, we rewrite the equation as:

   ${ \frac{(x - 1)(x - 2)}{x - 1} = x - 2 }$

   Factoring the quadratic expression allows us to identify potential simplifications and cancellations, which are essential techniques in algebra. The ability to quickly and accurately factor quadratic expressions is a valuable skill in mathematics. Mastering this technique can significantly speed up the problem-solving process.

2. Identify the restriction:

   Before we proceed with simplification, it is crucial to identify any restrictions on the variable. In this case, the denominator ${ x - 1 }$ cannot be zero, as division by zero is undefined. Therefore, we must state that ${ x \neq 1 }$. This restriction is vital and must be remembered throughout the solution process. Ignoring this restriction can lead to incorrect or extraneous solutions. The importance of identifying restrictions in rational equations cannot be overstated. It is a fundamental aspect of solving such equations correctly. When we identify this restriction early on, we ensure that our final solution does not include any values that would make the original equation undefined. This step is a cornerstone of sound algebraic practice.

3. Simplify the equation:

   We can cancel the common factor of ${ (x - 1) }$ from the numerator and the denominator, keeping in mind the restriction ${ x \neq 1 }$. This simplification yields:

   ${ x - 2 = x - 2 }$

   This simplification is valid only because we have acknowledged the restriction ${ x \neq 1 }$. Canceling common factors is a powerful technique in algebra, but it must be done with careful consideration of any restrictions. The simplified equation provides a clearer view of the relationship between the two sides of the equation. It highlights the fact that the two sides are identical, which has significant implications for the solution set.

4. Analyze the simplified equation:

   The simplified equation ${ x - 2 = x - 2 }$ is an identity, meaning it is true for all values of x. However, we must remember the restriction we identified earlier: ${ x \neq 1 }$. This restriction means that while the equation holds true for all other values of x, x cannot be equal to 1. The presence of the restriction is crucial in defining the final solution set. Analyzing the simplified equation in conjunction with the restriction is the key to finding the correct solution. The identity indicates that there are infinitely many solutions, but the restriction limits this set. Understanding how to interpret such equations and restrictions is a critical skill in advanced algebra and beyond.

Solution Set and Extraneous Solutions

From the simplified equation ${ x - 2 = x - 2 }$, it appears that any value of x would satisfy the equation. However, the crucial caveat is the restriction we identified earlier: ${ x \neq 1 }$. This restriction arises because the original equation contains a rational expression with ${ x - 1 }$ in the denominator. If we were to plug in ${ x = 1 }$ into the original equation, we would encounter division by zero, which is undefined. Therefore, even though ${ x = 1 }$ appears to be a solution based on the simplified equation, it is an extraneous solution. Extraneous solutions are values that satisfy a transformed equation but not the original equation. Identifying and excluding these solutions is a critical step in solving rational and radical equations. The solution set for the equation is all real numbers except ${ x = 1 }$. This can be expressed in various notations, such as interval notation or set-builder notation. Understanding the concept of extraneous solutions is fundamental in algebra. These solutions often arise when we perform operations such as squaring both sides of an equation or, as in this case, simplifying rational expressions. The process of solving equations is not complete until we have checked for and excluded any extraneous solutions. Failing to do so can lead to incorrect answers and a misunderstanding of the problem. The solution set represents the values of x that truly satisfy the original equation, taking into account any restrictions. Therefore, it is imperative to always verify solutions in the original equation, especially when dealing with rational or radical expressions.

Graphical Interpretation

To further understand the equation ${ \frac{x^2 - 3x + 2}{x - 1} = x - 2 }$, let’s consider its graphical interpretation. We can view the equation as the intersection of two functions:

${ f(x) = \frac{x^2 - 3x + 2}{x - 1} }$

and

${ g(x) = x - 2 }$

However, it's essential to note that the function ${ f(x) }$ has a removable discontinuity (a hole) at ${ x = 1 }$. This means that the graph of ${ f(x) }$ will look like the line ${ y = x - 2 }$ everywhere except at ${ x = 1 }$, where there will be a gap. Graphically, this hole represents the restriction we identified algebraically. When we plot both functions, we will see that the line ${ g(x) = x - 2 }$ overlaps with the graph of ${ f(x) }$ for all values of x except at ${ x = 1 }$. At ${ x = 1 }$, the function ${ f(x) }$ is undefined, creating a hole in the graph. The graphical representation provides a visual confirmation of our algebraic solution. It helps to reinforce the concept of extraneous solutions and the importance of considering restrictions. Visualizing the functions and their intersection points can enhance your understanding of the equation and its solution set. The graph of ${ f(x) }$ will appear as a straight line with a missing point, while the graph of ${ g(x) }$ is a continuous straight line. The intersection of these graphs will be all points on the line except the point where ${ x = 1 }$. This graphical perspective is a powerful tool for understanding and verifying algebraic solutions. It also highlights the connection between algebra and geometry, two fundamental branches of mathematics. By combining algebraic techniques with graphical analysis, we can gain a deeper and more complete understanding of mathematical problems.

Conclusion

In conclusion, solving the equation ${ \frac{x^2 - 3x + 2}{x - 1} = x - 2 }$ algebraically requires careful consideration of factoring, simplification, and, most importantly, restrictions. The initial step of factoring the numerator to ${ (x - 1)(x - 2) }$ sets the stage for simplification, but the crucial aspect is recognizing that ${ x \neq 1 }$ due to the original denominator. Simplifying the equation leads to an identity, ${ x - 2 = x - 2 }$, which is true for all x, but the restriction means that ${ x = 1 }$ must be excluded from the solution set. This exclusion highlights the importance of identifying and handling extraneous solutions. The solution set, therefore, includes all real numbers except 1. Graphically, the equation represents the intersection of two functions, where one function has a removable discontinuity at ${ x = 1 }$. This graphical interpretation reinforces the algebraic solution, showing a visual representation of why ${ x = 1 }$ is not a valid solution. The process of solving this equation underscores several key concepts in algebra, including factoring, simplifying rational expressions, identifying restrictions, and understanding the nature of extraneous solutions. These concepts are fundamental to solving a wide range of algebraic problems and are essential for success in higher-level mathematics. Furthermore, the connection between the algebraic and graphical approaches demonstrates the power of using multiple perspectives to understand mathematical concepts. By combining algebraic manipulation with graphical analysis, we can gain a more comprehensive understanding of the problem and its solution. This holistic approach not only enhances problem-solving skills but also fosters a deeper appreciation for the interconnectedness of different areas of mathematics. Thus, mastering the techniques discussed in this article will undoubtedly prove beneficial in your mathematical journey.