https://file.aiquickdraw.com/mathbot/file/17297129266717f7xq8i3.jpg

The equation given is:

\[
\frac{6}{x} + \frac{1}{x - 1} = -9
\]

Let's solve for \( x \).

### Step 1: Find a common denominator
The denominators are \( x \) and \( x - 1 \), so the least common denominator (LCD) is \( x(x - 1) \). Multiply both sides of the equation by this LCD to eliminate the fractions.

\[
x(x - 1) \left( \frac{6}{x} + \frac{1}{x - 1} \right) = x(x - 1)(-9)
\]

This simplifies to:

\[
6(x - 1) + 1(x) = -9x(x - 1)
\]

### Step 2: Simplify both sides

Expand both sides:

\[
6x - 6 + x = -9x^2 + 9x
\]

This simplifies to:

\[
7x - 6 = -9x^2 + 9x
\]

### Step 3: Rearrange the equation

Move all terms to one side:

\[
9x^2 - 2x - 6 = 0
\]

### Step 4: Solve the quadratic equation

Use the quadratic formula:

\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(9)(-6)}}{2(9)}
\]

Simplify the discriminant:

\[
x = \frac{2 \pm \sqrt{4 + 216}}{18} = \frac{2 \pm \sqrt{220}}{18}
\]

\[
x = \frac{2 \pm 2\sqrt{55}}{18}
\]

\[
x = \frac{1 \pm \sqrt{55}}{9}
\]

### Step 5: Solutions

The solutions are:

\[
x = \frac{1 + \sqrt{55}}{9}, \quad x = \frac{1 - \sqrt{55}}{9}
\]

These are the exact solutions. Would you like any further details on the solution steps?

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Here are some related questions:
1. How does the quadratic formula work in this case?
2. What are common denominators, and why are they useful?
3. How can you check the solutions to ensure they are correct?
4. What role does the discriminant play in the quadratic formula?
5. Can a quadratic equation have more than two real solutions?

**Tip:** When solving rational equations, multiplying by the least common denominator early can simplify the process significantly!

Learn how to solve the rational equation 6/x + 1/(x - 1) = -9 step by step, using algebraic techniques and the quadratic formula. This problem covers key algebraic concepts and is suitable for students in Grades 9-12.

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