Questions: Solve the following rational inequality (x-3)/(x^2-16)>0. Give your answer using interval notation.

To solve the rational inequality \(\frac{x-3}{x^2-16} > 0\), follow these steps:

1. Factor the denominator: \[ x^2 - 16 = (x - 4)(x + 4) \] So the inequality becomes: \[ \frac{x-3}{(x-4)(x+4)} > 0 \]

2. Identify critical points: The critical points are where the numerator or the denominator is zero. These points are: \[ x = 3, \quad x = 4, \quad x = -4 \]

3. Determine the sign of the expression in each interval: The critical points divide the number line into four intervals: \[ (-\infty, -4), \quad (-4, 3), \quad (3, 4), \quad (4, \infty) \]

   Test a point in each interval to determine the sign of the expression \(\frac{x-3}{(x-4)(x+4)}\):

   • For \(x \in (-\infty, -4)\), choose \(x = -5\): \[ \frac{-5-3}{(-5-4)(-5+4)} = \frac{-8}{(-9)(-1)} = \frac{-8}{9} < 0 \]
   • For \(x \in (-4, 3)\), choose \(x = 0\): \[ \frac{0-3}{(0-4)(0+4)} = \frac{-3}{(-4)(4)} = \frac{-3}{-16} = \frac{3}{16} > 0 \]
   • For \(x \in (3, 4)\), choose \(x = 3.5\): \[ \frac{3.5-3}{(3.5-4)(3.5+4)} = \frac{0.5}{(-0.5)(7.5)} = \frac{0.5}{-3.75} < 0 \]
   • For \(x \in (4, \infty)\), choose \(x = 5\): \[ \frac{5-3}{(5-4)(5+4)} = \frac{2}{(1)(9)} = \frac{2}{9} > 0 \]

4. Combine the intervals where the expression is positive: The expression \(\frac{x-3}{(x-4)(x+4)}\) is positive in the intervals \((-4, 3)\) and \((4, \infty)\).

5. Consider the critical points: Since the inequality is strict (\(>\)), the critical points themselves are not included in the solution.

6. Write the solution in interval notation: \[ (-4, 3) \cup (4, \infty) \]

So, the answer is: \[ (-4, 3) \cup (4, \infty) \]