Part BThe Image Shows The Rational Equation From Part A With An Incorrect Solution Process That A Student Performed. Explain The Error The Student Made, And Give The Correct Solution.Given The Rational Equation:$ \frac{x}{x-2} + \frac{1}{x-6} =

Introduction

Rational equations are a fundamental concept in algebra, and solving them requires a clear understanding of the rules and procedures involved. However, students often make mistakes when solving these equations, which can lead to incorrect solutions. In this article, we will examine a rational equation that a student attempted to solve, identify the error made, and provide the correct solution.

The Rational Equation

The rational equation given by the student is:

$$\frac{x}{x-2} + \frac{1}{x-6} = \frac{3}{x-2}$$

Error Analysis

Upon examining the student's work, we notice that they attempted to solve the equation by multiplying both sides by the least common denominator (LCD), which is $(x-2)(x-6)$. However, they made a critical mistake by not considering the restrictions on the values of $x$ that would make the denominators equal to zero.

Restrictions on the Denominators

To solve the equation, we need to consider the restrictions on the values of $x$ that would make the denominators equal to zero. In this case, the denominators are $(x-2)$ and $(x-6)$. Therefore, we must exclude the values $x=2$ and $x=6$ from the solution set.

Correct Solution

To solve the equation, we will follow the correct procedure:

1. Multiply both sides by the LCD: Multiply both sides of the equation by the least common denominator, which is $(x-2)(x-6)$.
2. Distribute the LCD: Distribute the LCD to each term in the equation.
3. Combine like terms: Combine like terms on the left-hand side of the equation.
4. Solve for x: Solve for $x$ by setting the numerator equal to zero and factoring the quadratic expression.

Step 1: Multiply both sides by the LCD

Multiply both sides of the equation by the least common denominator, $(x-2)(x-6)$:

$$\frac{x}{x-2} + \frac{1}{x-6} = \frac{3}{x-2}$$

$$(x-2)(x-6)\left(\frac{x}{x-2} + \frac{1}{x-6}\right) = (x-2)(x-6)\left(\frac{3}{x-2}\right)$$

Step 2: Distribute the LCD

Distribute the LCD to each term in the equation:

$$x(x-6) + (x-2) = 3(x-6)$$

Step 3: Combine like terms

Combine like terms on the left-hand side of the equation:

$$x^2 - 6x + x - 2 = 3x - 18$$

$$x^2 - 5x - 2 = 3x - 18$$

Step 4: Solve for x

Solve for $x$ by setting the numerator equal to zero and factoring the quadratic expression:

$$x^2 - 5x - 2 = 3x - 18$$

$$x^2 - 8x + 16 = 0$$

$$(x - 4)^2 = 0$$

$$x - 4 = 0$$

$$x = 4$$

Conclusion

In this article, we examined a rational equation that a student attempted to solve and identified the error made. We then provided the correct solution by following the correct procedure, including multiplying both sides by the least common denominator, distributing the LCD, combining like terms, and solving for $x$. The correct solution is $x = 4$, which is the only solution to the equation.

Final Answer

The final answer is $\boxed{4}$.