Thank you so much ### Solving Rational Inequalities and Graphing the Solution Set#### Problem Statement:Solve the following rational inequality and graph the solution set on a real number line:\[ \frac{x + 1}{x + 5} < 2 \]#### Solution Steps:1. **Rewrite the Inequality**: First, rewrite the inequality in a standard form by bringing all terms to one side. \[ \frac{x + 1}{x + 5} - 2 < 0 \]2. **Combine the Terms**: Combine the terms into a single fraction. \[ \frac{x + 1 - 2(x + 5)}{x + 5} < 0 \] Simplify the numerator: \[ \frac{x + 1 - 2x - 10}{x + 5} < 0 \] \[ \frac{-x - 9}{x + 5} < 0 \]3. **Determine Critical Points**: Find the values that make the numerator and denominator zero. - Numerator: \(-x - 9 = 0\) ⟹ \( x = -9 \) - Denominator: \(x + 5 = 0\) ⟹ \( x = -5 \)4. **Test Intervals**: The critical points divide the number line into intervals. Test each interval to determine where the inequality holds true: - Interval 1: \( (-\infty, -9) \) - Interval 2: \( (-9, -5) \) - Interval 3: \( (-5, \infty) \) Select test points for each interval: - For \( (-\infty, -9) \), test \( x = -10 \) - For \( (-9, -5) \), test \( x = -7 \) - For \( (-5, \infty) \), test \( x = 0 \) Evaluate the inequality \( \frac{-x - 9}{x + 5} < 0 \) for each test point.5. **Graph the Solution**: Graph the solution set on a real number line, marking the intervals where the inequality is satisfied. Note that the critical points \( x=-9 \) and \( x=-5 \) will not

Name: Thank you so much ### Solving Rational Inequalities and Graphing the S
Uploaded: 2024-03-05T11:52:21.000Z
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Description: Thank you so much ### Solving Rational Inequalities and Graphing the Solution Set#### Problem Statement:Solve the following rational inequality and graph the solution set on a real number line:\[ \frac{x + 1}{x + 5} < 2 \]#### Solution Steps:1. **Rew