Solve for all values of x, .*x*.

start fraction, x, minus, 9, divided by, x, plus, 1, end fraction, equals, start fraction, minus, 4, divided by, x, end fraction

*x*+1*x*−9​=

*x*−4​

Let's solve the equation:

Given:
\[
\frac{x - 9}{x + 1} = \frac{-4}{x}
\]

### Step 1: Cross-multiply
We start by cross-multiplying to eliminate the fractions:
\[
(x - 9) \cdot x = -4 \cdot (x + 1)
\]

### Step 2: Expand both sides
Next, expand both sides of the equation:
\[
x(x - 9) = -4(x + 1)
\]
\[
x^2 - 9x = -4x - 4
\]

### Step 3: Move all terms to one side of the equation
Move all the terms to one side of the equation to set it equal to zero:
\[
x^2 - 9x + 4x + 4 = 0
\]
Simplify the equation:
\[
x^2 - 5x + 4 = 0
\]

### Step 4: Factor the quadratic equation
Now, we factor the quadratic equation:
\[
x^2 - 5x + 4 = (x - 4)(x - 1) = 0
\]

### Step 5: Solve for \(x\)
Set each factor equal to zero:
\[
x - 4 = 0 \quad \text{or} \quad x - 1 = 0
\]
\[
x = 4 \quad \text{or} \quad x = 1
\]

### Step 6: Check for any extraneous solutions
We need to check if these solutions satisfy the original equation:
- For \(x = 4\):
  \[
  \frac{4 - 9}{4 + 1} = \frac{-5}{5} = -1 \quad \text{and} \quad \frac{-4}{4} = -1
  \]
  Both sides are equal, so \(x = 4\) is a valid solution.

- For \(x = 1\):
  \[
  \frac{1 - 9}{1 + 1} = \frac{-8}{2} = -4 \quad \text{and} \quad \frac{-4}{1} = -4
  \]
  Both sides are equal, so \(x = 1\) is also a valid solution.

### Final Answer:
The solutions are \( \boxed{x = 4 \text{ and } x = 1} \).

Would you like more details, or do you have any other questions?

Here are 8 related questions to consider:
1. How do you solve quadratic equations by completing the square?
2. What is the difference between a factorable quadratic and one that requires the quadratic formula?
3. How do you solve equations with absolute value?
4. What are the properties of rational functions?
5. How can you determine the number of solutions a quadratic equation has?
6. What is the role of the discriminant in determining the nature of solutions for a quadratic equation?
7. How do you solve equations involving square roots?
8. Can all quadratic equations be factored easily?

**Tip:** Always remember to verify your solutions by substituting them back into the original equation, especially when dealing with rational expressions, to ensure they are not extraneous.

Learn how to solve the rational equation (x - 9) / (x + 1) = (-4) / x with a detailed step-by-step solution. This problem covers algebraic concepts including rational expressions and cross-multiplication. Suitable for students in Grades 10-12.

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