Solve this rational inequality

x-1	/x*2-9 < 0

To solve the rational inequality:

\[
\frac{x - 1}{x^2 - 9} < 0
\]

### Step 1: Factor the denominator.
The denominator \(x^2 - 9\) can be factored as:

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

Thus, the inequality becomes:

\[
\frac{x - 1}{(x - 3)(x + 3)} < 0
\]

### Step 2: Find the critical points.
The critical points are the values of \(x\) that make the numerator or denominator equal to zero. Set the numerator and denominator equal to zero:

1. \(x - 1 = 0 \Rightarrow x = 1\)
2. \(x - 3 = 0 \Rightarrow x = 3\)
3. \(x + 3 = 0 \Rightarrow x = -3\)

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

### Step 3: Determine the sign of the expression in each interval.
The critical points divide the real number line into four intervals:

1. \( (-\infty, -3) \)
2. \( (-3, 1) \)
3. \( (1, 3) \)
4. \( (3, \infty) \)

We will now test the sign of the expression \(\frac{x - 1}{(x - 3)(x + 3)}\) in each interval.

1. **Interval \( (-\infty, -3) \):** Pick \(x = -4\)
   \[
   \frac{-4 - 1}{(-4 - 3)(-4 + 3)} = \frac{-5}{(-7)(-1)} = \frac{-5}{7} < 0
   \]
   The expression is negative in this interval.

2. **Interval \( (-3, 1) \):** Pick \(x = 0\)
   \[
   \frac{0 - 1}{(0 - 3)(0 + 3)} = \frac{-1}{(-3)(3)} = \frac{-1}{-9} = \frac{1}{9} > 0
   \]
   The expression is positive in this interval.

3. **Interval \( (1, 3) \):** Pick \(x = 2\)
   \[
   \frac{2 - 1}{(2 - 3)(2 + 3)} = \frac{1}{(-1)(5)} = \frac{-1}{5} < 0
   \]
   The expression is negative in this interval.

4. **Interval \( (3, \infty) \):** Pick \(x = 4\)
   \[
   \frac{4 - 1}{(4 - 3)(4 + 3)} = \frac{3}{(1)(7)} = \frac{3}{7} > 0
   \]
   The expression is positive in this interval.

### Step 4: Write the solution.
We are looking for where the expression is less than zero, which happens in the intervals where the expression is negative.

The solution is:

\[
x \in (-\infty, -3) \cup (1, 3)
\]

### Step 5: Exclude critical points where the expression is undefined.
At \(x = -3\) and \(x = 3\), the denominator is zero, so these points are excluded from the solution.

The final solution is:

\[
x \in (-\infty, -3) \cup (1, 3)
\]

---

Would you like further details on any of the steps? Here are some related questions:

1. How do you solve rational inequalities in general?
2. What happens if the inequality were \(\leq\) instead of \(<\)?
3. How do you handle cases where the numerator has higher degree than the denominator?
4. What if the inequality were \(\frac{x - 1}{x^2 - 9} > 0\)?
5. How does the critical point method help in finding the solution intervals?

**Tip:** Always check the sign of the expression in each interval between the critical points; this is key to solving rational inequalities.

Learn how to solve the rational inequality (x - 1) / (x^2 - 9) < 0 with a detailed step-by-step solution. This problem covers key concepts in inequalities and rational functions, including factorization of quadratic expressions. Suitable for students in Grades 10-12.

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