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 step by step:

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

### Step 1: Cross-multiply
To eliminate the fractions, we'll cross-multiply:
\[
(x - 9) \cdot x = -4 \cdot (x + 1)
\]

### Step 2: Expand both sides
Expand the products on both sides:
\[
x^2 - 9x = -4x - 4
\]

### Step 3: Move all terms to one side of the equation
Add \(4x\) and \(4\) to both sides:
\[
x^2 - 9x + 4x + 4 = 0
\]
Simplify the equation:
\[
x^2 - 5x + 4 = 0
\]

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

### 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 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 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 using the quadratic formula?
2. What are extraneous solutions and how do they arise?
3. How do you factor quadratic equations?
4. Can cross-multiplication always be used in solving rational equations?
5. How do you check if a solution is extraneous?
6. What are the different methods to solve quadratic equations?
7. How can you solve equations involving higher-degree polynomials?
8. What is the significance of the discriminant in a quadratic equation?

**Tip:** Always check your solutions in the original equation, especially when dealing with rational equations, to ensure no extraneous solutions are included.

Learn how to solve the rational equation (x - 9) / (x + 1) = (-4) / x step-by-step with detailed explanations. This problem involves rational equations, cross-multiplication, and quadratic equation concepts. Suitable for students in Grades 9-12.

Math Problem Statement

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

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