find all vertical asymptotes for the equation y=x+4/x^2-1

To find the vertical asymptotes for the equation \( y = \frac{x + 4}{x^2 - 1} \), we need to identify the values of \( x \) that make the denominator equal to zero, since division by zero is undefined and typically indicates a vertical asymptote.

### Step 1: Set the denominator equal to zero.
\[
x^2 - 1 = 0
\]

### Step 2: Solve for \( x \).
This is a difference of squares, which can be factored as:
\[
(x - 1)(x + 1) = 0
\]
Thus, \( x - 1 = 0 \) or \( x + 1 = 0 \).

### Step 3: Find the values of \( x \).
\[
x = 1 \quad \text{or} \quad x = -1
\]

### Conclusion:
The vertical asymptotes occur at \( x = 1 \) and \( x = -1 \).

Would you like more details on this solution or have any other questions?

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Here are five related questions to expand your understanding:
1. How do you find the horizontal asymptotes of a rational function?
2. What is the behavior of the function as it approaches the vertical asymptotes?
3. How can you find the slant (oblique) asymptotes of a rational function?
4. How does the presence of vertical asymptotes affect the graph of the function?
5. Can a function have more than two vertical asymptotes?

**Tip:** Always remember to check for holes in the graph of a rational function, which occur when both the numerator and denominator have a common factor that can be canceled out.

Discover how to find the vertical asymptotes for the rational function y=(x+4)/(x^2-1). This detailed explanation covers key concepts such as the difference of squares and vertical asymptotes, suitable for students in Grades 9-12.

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