https://file.aiquickdraw.com/mathbot/file/1725899926624gtxqzy5z.jpg

The equation in the image is a rational equation given by:

\[
\frac{x}{x - 1} = \frac{6}{x + 1}
\]

Let's solve this equation step-by-step:

### Step 1: Cross-Multiply
To eliminate the fractions, cross-multiply:

\[
x \cdot (x + 1) = 6 \cdot (x - 1)
\]

### Step 2: Expand Both Sides
Expand both sides of the equation:

\[
x^2 + x = 6x - 6
\]

### Step 3: Move All Terms to One Side
Move all terms to one side of the equation to form a quadratic equation:

\[
x^2 + x - 6x + 6 = 0
\]

Simplify:

\[
x^2 - 5x + 6 = 0
\]

### Step 4: Factor the Quadratic Equation
Factor the quadratic equation:

\[
(x - 2)(x - 3) = 0
\]

### Step 5: Solve for \(x\)
Set each factor equal to zero:

\[
x - 2 = 0 \quad \text{or} \quad x - 3 = 0
\]

So, the solutions are:

\[
x = 2 \quad \text{or} \quad x = 3
\]

### Step 6: Check for Extraneous Solutions
Ensure the solutions do not make the original denominators zero:

- For \(x = 2\): The denominators are \(2 - 1 = 1\) and \(2 + 1 = 3\). Both are non-zero.
- For \(x = 3\): The denominators are \(3 - 1 = 2\) and \(3 + 1 = 4\). Both are non-zero.

Both solutions are valid.

**Final Solutions: \(x = 2\) and \(x = 3\)**

Would you like further details or have any questions about this problem?

### Related Questions:
1. How do you solve rational equations with more complex numerators and denominators?
2. What are the common methods to check for extraneous solutions in rational equations?
3. Can rational equations have no solutions? If so, how can you identify such cases?
4. How would you graph the functions involved in this rational equation?
5. What are the applications of rational equations in real-world problems?

**Tip:** Always check for extraneous solutions when solving rational equations, as some solutions may not be valid in the original equation due to restrictions on the variable.

Solve the rational equation: \(\frac{x}{x-1} = \frac{6}{x+1}\)

Learn how to solve the rational equation \(\frac{x}{x-1} = \frac{6}{x+1}\) with a detailed step-by-step solution. This problem covers key algebraic concepts like cross-multiplication and factoring quadratics. Suitable for students in Grades 9-11.

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