Solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval notation. \[ \frac{x+3}{x+5}<2 \] Solve the inequality. What is the solution set? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is (Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) B. The solution set is the empty set.

To solve the rational inequality \[ \frac{x+3}{x+5} < 2, \] we will follow these steps: 1. **Rearrange the inequality**: We want to bring all terms to one side of the inequality. We can do this by subtracting 2 from both sides: \[ \frac{x+3}{x+5} - 2 < 0. \] 2. **Combine the fractions**: To combine the fractions, we need a common denominator. The common denominator is \(x + 5\): \[ \frac{x+3 - 2(x+5)}{x+5} < 0. \] 3. **Simplify the numerator**: Distributing the -2 gives us: \[ \frac{x + 3 - 2x - 10}{x + 5} < 0, \] which simplifies to: \[ \frac{-x - 7}{x + 5} < 0. \] 4. **Factor the numerator**: We can factor out a -1 from the numerator: \[ \frac{-(x + 7)}{x + 5} < 0. \] 5. **Determine the critical points**: The critical points occur when the numerator or denominator is zero: - The numerator \( -(x + 7) = 0 \) gives \( x = -7 \). - The denominator \( x + 5 = 0 \) gives \( x = -5 \). Thus, the critical points are \( x = -7 \) and \( x = -5 \). 6. **Test intervals**: We will test the intervals determined by the critical points: - Interval 1: \( (-\infty, -7) \) - Interval 2: \( (-7, -5) \) - Interval 3: \( (-5, \infty) \) We will choose test points from each interval: - For \( x = -8 \) in \( (-\infty, -7) \): \[ \frac{-(-8 + 7)}{-8 + 5} = \frac{-(-1)}{-3} = \frac{1}{-3} < 0 \quad \text{(True)} \] - For \( x = -6 \) in \( (-7, -5) \): \[ \frac{-(-6 + 7)}{-6 + 5} = \frac{-1}{-1} = 1 > 0 \quad \text{(False)} \] - For \( x = 0 \) in \( (-5, \infty) \): \[ \frac{-(0 + 7)}{0 + 5} = \frac{-7}{5} < 0 \quad \text{(True)} \] 7. **Combine the results**: The inequality is satisfied in the intervals \( (-\infty, -7) \) and \( (-5, \infty) \). 8. **Express in interval notation**: The solution set is: \[ (-\infty, -7) \cup (-5, \infty). \] Thus, the final answer is: A. The solution set is \((- \infty, -7) \cup (-5, \infty)\).