We are given the equation:

\[
\frac{1}{x-1} + \frac{x}{x+17} = \frac{18}{x^2 + 16x - 17}
\]

The goal is to simplify and solve for \( x \).

### Step 1: Factor the denominator on the right-hand side
We start by factoring the quadratic expression on the right-hand side:

\[
x^2 + 16x - 17 = (x - 1)(x + 17)
\]

Thus, the equation becomes:

\[
\frac{1}{x-1} + \frac{x}{x+17} = \frac{18}{(x-1)(x+17)}
\]

### Step 2: Combine the left-hand side into a single fraction
To combine the two fractions on the left-hand side, we need a common denominator, which will be \( (x-1)(x+17) \):

\[
\frac{1}{x-1} = \frac{x+17}{(x-1)(x+17)}, \quad \frac{x}{x+17} = \frac{x(x-1)}{(x-1)(x+17)}
\]

Thus, the left-hand side becomes:

\[
\frac{x+17}{(x-1)(x+17)} + \frac{x(x-1)}{(x-1)(x+17)} = \frac{x+17 + x(x-1)}{(x-1)(x+17)}
\]

### Step 3: Simplify the numerator
Now, simplify the expression in the numerator:

\[
x + 17 + x(x-1) = x + 17 + x^2 - x = x^2 + 17
\]

So, the left-hand side becomes:

\[
\frac{x^2 + 17}{(x-1)(x+17)}
\]

### Step 4: Set the two sides equal to each other
Now the equation is:

\[
\frac{x^2 + 17}{(x-1)(x+17)} = \frac{18}{(x-1)(x+17)}
\]

### Step 5: Eliminate the denominators
Since the denominators are the same, we can cancel them, leaving us with:

\[
x^2 + 17 = 18
\]

### Step 6: Solve for \( x \)
Now, subtract 17 from both sides:

\[
x^2 = 1
\]

Take the square root of both sides:

\[
x = \pm 1
\]

### Step 7: Check for extraneous solutions
We must check if \( x = 1 \) is a valid solution by substituting it back into the original equation. If \( x = 1 \), the term \( \frac{1}{x-1} \) is undefined because the denominator would be zero. Therefore, \( x = 1 \) is an extraneous solution.

Thus, the only valid solution is:

\[
x = -1
\]

### Final Answer:
The solution to the equation is \( x = -1 \).

---

Would you like any more details or have any questions?

### Related Questions:
1. How can you identify extraneous solutions in equations involving rational expressions?
2. What are the steps for factoring quadratic expressions like \(x^2 + 16x - 17\)?
3. How do you find the least common denominator when combining fractions?
4. What methods are used to solve quadratic equations?
5. How do you verify solutions of rational equations?

### Tip:
Always check for extraneous solutions when dealing with rational equations, as certain values may make the denominator zero, invalidating the solution.

Learn how to solve the rational equation (1/(x-1)) + (x/(x+17)) = 18/(x^2 + 16x - 17) with a step-by-step solution. This problem includes key algebraic concepts such as factoring quadratic expressions, combining rational fractions, and identifying extraneous solutions. Suitable for students in Grades 9-11.

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