Solving Rational Inequalities: Testing Intervals with a Constant

Algebraic Manipulation

• 💡 The first step is to manipulate the inequality to have zero on one side and a single rational expression on the other.
• 🎯 This involves subtracting the constant from both sides and finding a common denominator to combine terms.
• 🔑 The inequality 16 / (x^2 - 2x) - 2 < 0 is transformed into (-2x^2 + 4x + 16) / (x^2 - 2x) < 0.

Factoring and Critical Points

• 🚀 The rational expression is then factored to identify critical points where the numerator or denominator equals zero.
• 🔍 The numerator -2x^2 + 4x + 16 factors into -2(x - 4)(x + 2), yielding critical points at x = 4 and x = -2.
• ⚠️ The denominator x^2 - 2x factors into x(x - 2), yielding critical points at x = 0 and x = 2.
• ✅ These critical points (-2, 0, 2, 4) divide the number line into intervals where the sign of the expression may change.

Interval Testing

• 🔬 Each interval defined by the critical points is tested to determine if the inequality f(x) < 0 holds true.
• 🧪 For x < -2 (e.g., x = -3), the expression is negative, satisfying the inequality.
• ❌ For -2 < x < 0 (e.g., x = -1), the expression is positive, not satisfying the inequality.
• ✅ For 0 < x < 2 (e.g., x = 1), the expression is negative, satisfying the inequality.
• ❌ For 2 < x < 4 (e.g., x = 3), the expression is positive, not satisfying the inequality.
• ✅ For x > 4 (e.g., x = 5), the expression is negative, satisfying the inequality.

Solution Set

• 📈 The solution set includes all intervals where the rational inequality is strictly less than zero.
• 🧩 The solution is the union of the intervals x < -2, 0 < x < 2, and x > 4.
• 💬 In interval notation, this is represented as (-∞, -2) ∪ (0, 2) ∪ (4, ∞).