Hey, GPT-40. I've got the equation: y=1/(x^2-1). I am looking for x. Do you mind helping me out?

Certainly! You want to solve the equation \( y = \frac{1}{x^2 - 1} \) for \( x \).

Here's how to approach it:

### Step 1: Express the equation in terms of \( x^2 - 1 \)
Multiply both sides by \( x^2 - 1 \) to get rid of the fraction:
\[
y(x^2 - 1) = 1
\]

### Step 2: Expand and rearrange the equation
Distribute \( y \) on the left side:
\[
yx^2 - y = 1
\]
Next, move the \( -y \) term to the right side:
\[
yx^2 = y + 1
\]

### Step 3: Solve for \( x^2 \)
Divide both sides by \( y \) (assuming \( y \neq 0 \)):
\[
x^2 = \frac{y + 1}{y}
\]

### Step 4: Solve for \( x \)
Now, take the square root of both sides:
\[
x = \pm \sqrt{\frac{y + 1}{y}}
\]

So, the solution for \( x \) is:
\[
x = \pm \sqrt{\frac{y + 1}{y}}
\]

### Additional Consideration:
- The equation \( y = \frac{1}{x^2 - 1} \) is undefined when \( x^2 - 1 = 0 \), which occurs at \( x = \pm 1 \). Therefore, the solution \( x = \pm 1 \) must be excluded.

Would you like more details on any of these steps?

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Here are five related questions:
1. What are the critical points of the function \( y = \frac{1}{x^2 - 1} \)?
2. How does the graph of \( y = \frac{1}{x^2 - 1} \) look, and where are its asymptotes?
3. What happens to the value of \( x \) as \( y \) approaches infinity or zero?
4. How would you solve the equation if \( y \) were a function of another variable, such as \( y = 2x + 3 \)?
5. Can the equation \( y = \frac{1}{x^2 - 1} \) have complex solutions for \( x \)?

**Tip:** When dealing with equations that have square roots, always consider both the positive and negative roots and check for any values that may cause the original equation to be undefined.

Learn how to solve the equation y = 1/(x^2 - 1) for x step-by-step. Understand the principles of quadratic equations and function analysis with detailed explanations. Suitable for high school students.

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