Solving Rational Inequalities: Testing Intervals with Rational Expressions

Rewriting the Inequality

• 🎯 The first step is to rewrite the inequality so that one side is zero, allowing for interval testing.
• 💡 This involves algebraic manipulation, such as subtracting terms from both sides to achieve the zero.

Simplifying the Rational Expression

• 🛠️ Combine terms into a single rational expression by finding a common denominator.
• ⚠️ Be mindful of removable discontinuities (e.g., where a factor cancels from numerator and denominator), noting these as restrictions on the domain (e.g., x cannot equal -2).
• 🧩 After combining terms, simplify the numerator and denominator. For example, x^2 terms may cancel out.

Identifying Critical Points

• 🔍 Critical points for sign changes are where the numerator equals zero or where there are non-removable discontinuities (where the denominator equals zero).
• 📌 In this case, the critical points are x = 2 (from the numerator 2x - 4) and x = -3 (from the denominator x + 3).

Testing Intervals

• 📊 Test intervals defined by the critical points to determine where the rational expression is greater than or equal to zero.
• 🧪 For x < -3 (e.g., x = -4), the expression is positive.
• 🧪 For -3 < x < 2 (e.g., x = 0), the expression is negative.
• 🧪 For x > 2 (e.g., x = 3), the expression is positive.

Solution Set

• ✅ The solution set includes intervals where the expression is positive or zero.
• 📈 This includes x < -3 and x ≥ 2.
• 🔗 The solution can be expressed in interval notation as (-∞, -3) U [2, ∞), remembering the domain restriction x ≠ -2.