# How do you solve: #(x^2-9)/(x^2-1) < 0#?

Your inequality looks like this

More specifically, you need to have

Now, in order for this inequality to be true, you need to have

or

For the fist set of conditions to be true, you need to have

For the second set of conditions, you need to have

graph{(x^2 - 9)/(x^2 - 1) [-18.02, 18.01, -9.01, 9.01]}

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To solve the inequality (\frac{x^2 - 9}{x^2 - 1} < 0), we first need to find the critical points where the expression equals zero or is undefined. The critical points occur where the numerator or denominator equals zero.

The numerator (x^2 - 9) equals zero when (x = \pm 3), and the denominator (x^2 - 1) equals zero when (x = \pm 1).

So, the critical points are (x = -3), (x = -1), (x = 1), and (x = 3).

Next, we need to test each interval created by these critical points to determine where the expression (\frac{x^2 - 9}{x^2 - 1}) is negative. We can do this by choosing test points in each interval and evaluating the expression.

- Test a point in the interval ((- \infty, -3)), for example, (x = -4): (\frac{(-4)^2 - 9}{(-4)^2 - 1} > 0)
- Test a point in the interval ((-3, -1)), for example, (x = -2): (\frac{(-2)^2 - 9}{(-2)^2 - 1} < 0)
- Test a point in the interval ((-1, 1)), for example, (x = 0): (\frac{(0)^2 - 9}{(0)^2 - 1} > 0)
- Test a point in the interval ((1, 3)), for example, (x = 2): (\frac{(2)^2 - 9}{(2)^2 - 1} < 0)
- Test a point in the interval ((3, \infty)), for example, (x = 4): (\frac{(4)^2 - 9}{(4)^2 - 1} > 0)

So, the solution to the inequality (\frac{x^2 - 9}{x^2 - 1} < 0) is (x \in (-3, -1) \cup (1, 3)).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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