Solving Rational Inequalities: Testing Intervals for f(x) >= 0 | Precalculus

Identifying Critical Points

• 🔑 The first step in solving rational inequalities is to find the critical points where the function f(x) could transition signs.
• 🎯 These points include where f(x) = 0 (numerator is zero) and where f(x) has a vertical asymptote (denominator is zero).
• 💡 For the given f(x), the points where f(x) = 0 are x = 3 and x = -2.
• ⚠️ The vertical asymptotes occur at x = -1/2 (from 2x + 1 = 0) and x = 5 (from x - 5 = 0).

Analyzing Intervals on a Number Line

• 📈 A number line is used to visualize these critical points: -2, -1/2, 3, and 5.
• 🚫 The points where the denominator is zero (x = -1/2 and x = 5) are excluded from the solution set because they are restrictions on the domain.
• ✅ The points where the numerator is zero (x = -2 and x = 3) are included in the solution set because the inequality is "greater than or equal to zero."

Testing Intervals for Sign Changes

• 🔬 To determine where f(x) >= 0, test a value within each interval created by the critical points.
• 🧪 Testing x = -3 (interval (-∞, -2)) yields a positive result, so this interval is included.
• 🧪 Testing x = -1 (interval (-2, -1/2)) yields a negative result, so this interval is excluded.
• 🧪 Testing x = 0 (interval (-1/2, 3)) yields a positive result (specifically, 6/5), so this interval is included.
• 🧪 Testing x = 4 (interval (3, 5)) yields a negative result, so this interval is excluded.
• 🧪 Testing x = 6 (interval (5, ∞)) yields a positive result, so this interval is included.

Final Solution Set

• 🧩 The intervals where f(x) is greater than or equal to zero are (-∞, -2], (-1/2, 3], and (5, ∞).
• 🔗 Combining these, the solution set is the union of these intervals.