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We are asked to solve the rational inequality:

\[
\frac{x - 6}{x^2 - 49} > 0
\]

### Step 1: Factor the denominator
The denominator \(x^2 - 49\) is a difference of squares, which can be factored as:

\[
x^2 - 49 = (x - 7)(x + 7)
\]

So the inequality becomes:

\[
\frac{x - 6}{(x - 7)(x + 7)} > 0
\]

### Step 2: Identify critical points
The critical points are where the expression equals zero or is undefined. These occur when either the numerator or denominator is zero.

- \(x - 6 = 0 \implies x = 6\)
- \(x - 7 = 0 \implies x = 7\)
- \(x + 7 = 0 \implies x = -7\)

Thus, the critical points are \(x = 6\), \(x = 7\), and \(x = -7\). These points divide the number line into intervals.

### Step 3: Analyze sign changes in each interval
We will test the sign of the expression in each of the following intervals:

1. \( (-\infty, -7) \)
2. \( (-7, 6) \)
3. \( (6, 7) \)
4. \( (7, \infty) \)

Pick test points from each interval to check the sign of the expression \( \frac{x - 6}{(x - 7)(x + 7)} \).

- For \(x = -8\) (in \( (-\infty, -7) \)):
  \[
  \frac{-8 - 6}{(-8 - 7)(-8 + 7)} = \frac{-14}{(-15)(-1)} = \frac{-14}{15} < 0
  \]

- For \(x = 0\) (in \( (-7, 6) \)):
  \[
  \frac{0 - 6}{(0 - 7)(0 + 7)} = \frac{-6}{(-7)(7)} = \frac{-6}{-49} > 0
  \]

- For \(x = 6.5\) (in \( (6, 7) \)):
  \[
  \frac{6.5 - 6}{(6.5 - 7)(6.5 + 7)} = \frac{0.5}{(0.5)(13.5)} = \frac{0.5}{6.75} > 0
  \]

- For \(x = 8\) (in \( (7, \infty) \)):
  \[
  \frac{8 - 6}{(8 - 7)(8 + 7)} = \frac{2}{(1)(15)} = \frac{2}{15} > 0
  \]

### Step 4: Include or exclude critical points
- At \(x = 6\), the numerator is zero, so the entire expression is zero. We do not include this point in the solution, as we are looking for where the expression is strictly greater than zero.
- At \(x = 7\) and \(x = -7\), the denominator is zero, which makes the expression undefined. These points are also excluded from the solution.

### Step 5: Write the solution in interval notation
The expression is positive in the intervals \( (-7, 6) \), \( (6, 7) \), and \( (7, \infty) \).

Thus, the solution in interval notation is:

\[
(-7, 6) \cup (6, 7) \cup (7, \infty)
\]

Let me know if you need more details or further clarification!

### Related Questions:
1. How do you solve rational inequalities that have multiple factors in the denominator?
2. What is the significance of critical points in determining the solution of an inequality?
3. How can a sign chart help in solving inequalities involving rational expressions?
4. What are some common mistakes to avoid when solving rational inequalities?
5. Why are the points where the denominator is zero excluded from the solution?

### Tip:
Always double-check whether critical points should be included in your final solution by considering whether the inequality is strict (\(>\) or \(<\)) or non-strict (\(\geq\) or \(\leq\)).

Solve the following rational inequality: (x - 6) / (x^2 - 49) > 0. State your answer using interval notation. Use U for union and oo for infinity.

Learn how to solve the rational inequality (x - 6) / (x^2 - 49) > 0. This solution breaks down the inequality step by step, including factoring, identifying critical points, and analyzing sign changes across intervals. Suitable for high school students in Grades 10-12.

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