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The given equation to solve is:

\[
8 + \frac{3}{x - 1} = -\frac{7}{x - 2}
\]

Let's solve for \( x \).

### Step 1: Move terms to simplify the equation
First, subtract 8 from both sides to isolate the fractions:

\[
\frac{3}{x - 1} = -8 - \frac{7}{x - 2}
\]

### Step 2: Combine the right-hand side
Now, combine the terms on the right-hand side by getting a common denominator:

\[
\frac{3}{x - 1} = -\frac{8(x - 2) + 7}{x - 2}
\]

Simplify the numerator:

\[
\frac{3}{x - 1} = -\frac{8x - 16 + 7}{x - 2} = -\frac{8x - 9}{x - 2}
\]

### Step 3: Cross-multiply
Now, cross-multiply to solve for \( x \):

\[
3(x - 2) = -(8x - 9)(x - 1)
\]

### Step 4: Expand both sides
Expand both sides of the equation:

\[
3x - 6 = -(8x^2 - 8x - 9x + 9)
\]
\[
3x - 6 = -(8x^2 - 17x + 9)
\]
\[
3x - 6 = -8x^2 + 17x - 9
\]

### Step 5: Move all terms to one side
Move all terms to the left-hand side to set the equation to zero:

\[
8x^2 + 3x - 17x - 6 + 9 = 0
\]
\[
8x^2 - 14x + 3 = 0
\]

### Step 6: Solve the quadratic equation
Now solve the quadratic equation \( 8x^2 - 14x + 3 = 0 \) using the quadratic formula:

\[
x = \frac{-(-14) \pm \sqrt{(-14)^2 - 4(8)(3)}}{2(8)}
\]
\[
x = \frac{14 \pm \sqrt{196 - 96}}{16}
\]
\[
x = \frac{14 \pm \sqrt{100}}{16}
\]
\[
x = \frac{14 \pm 10}{16}
\]

### Step 7: Calculate the solutions
Now, calculate the two possible solutions:

\[
x = \frac{14 + 10}{16} = \frac{24}{16} = \frac{3}{2}
\]
\[
x = \frac{14 - 10}{16} = \frac{4}{16} = \frac{1}{4}
\]

### Step 8: Check for restrictions
The original equation has restrictions on the values of \( x \), as the denominators \( x - 1 \) and \( x - 2 \) cannot be zero. Therefore, \( x \neq 1 \) and \( x \neq 2 \).

Since neither \( \frac{3}{2} \) nor \( \frac{1}{4} \) violates these restrictions, both are valid solutions.

### Final Answer:
The solutions are:

\[
x = \frac{3}{2}, \frac{1}{4}
\]

Would you like more details on any step or have any questions?

---

Here are 5 related questions to expand your understanding:

1. How can you solve quadratic equations using factoring?
2. What are the common mistakes when solving rational equations?
3. How do you simplify complex fractions?
4. How do restrictions in the domain affect the solutions of rational equations?
5. What is the quadratic formula, and when is it used?

**Tip:** Always check for restrictions in rational equations to avoid invalid solutions.

Solve for x.
8 + (3 / (x - 1)) = - (7 / (x - 2))
If there is more than one solution, separate them with commas. If there is no solution, click on "No solution".

Learn how to solve the rational equation 8 + (3 / (x - 1)) = - (7 / (x - 2)) with a step-by-step solution. This problem covers key algebraic concepts, including rational equations and the quadratic formula. Suitable for students in Grades 9-12.

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