1. Problem 226: Find how many integers satisfy $$\frac{(x+2)(x+3)}{x-2} \geq 0$$ and are less than 5. 2. Step 1: Identify critical points where numerator or denominator is zero: $$x=-3, x=-2, x=2$$. 3. Step 2: Determine sign intervals for $$\frac{(x+2)(x+3)}{x-2}$$: - For $$x<-3$$: numerator positive (both factors negative), denominator negative, fraction positive. - For $$-32$$: numerator positive, denominator positive, fraction positive. 4. Step 3: Include points where fraction is zero: $$x=-3, x=-2$$ (numerator zero), exclude $$x=2$$ (denominator zero). 5. Step 4: Solution set is $$(-\infty,-3] \cup [-2,2) \cup (2,\infty)$$ where fraction $$\geq 0$$. 6. Step 5: Find integers less than 5 in solution set: - Integers $$\leq -3$$: $$..., -4, -3$$ - Integers $$-2 \leq x < 2$$: $$-2, -1, 0, 1$$ - Integers $$2 < x < 5$$: $$3, 4$$ 7. Step 6: Combine and count integers less than 5: $$-4, -3, -2, -1, 0, 1, 3, 4$$ total 8 integers. 8. Step 7: Check options: none match 8, so re-examine domain at $$x=2$$ (excluded), so 2 not included. 9. Step 8: Verify if $$x=2$$ is excluded, so integers less than 5 satisfying inequality are $$-4, -3, -2, -1, 0, 1, 3, 4$$ (8 integers). 10. Since options do not include 8, check if question asks for integers strictly less than 5 (yes), so answer is 8. --- 11. Problem 227: Of 150 houses, 60% have air-conditioning (A), 50% sunporch (S), 30% pool (P), 5 have all three, 5 have none. Find number with exactly two amenities. 12. Step 1: Calculate counts: - $$|A|=0.6 \times 150=90$$ - $$|S|=0.5 \times 150=75$$ - $$|P|=0.3 \times 150=45$$ - $$|A \cap S \cap P|=5$$ - $$|\text{none}|=5$$ 13. Step 2: Total houses with at least one amenity: $$150 - 5 = 145$$. 14. Step 3: Use inclusion-exclusion: $$|A \cup S \cup P| = |A| + |S| + |P| - |A \cap S| - |A \cap P| - |S \cap P| + |A \cap S \cap P| = 145$$ 15. Step 4: Let $$x = |A \cap S| + |A \cap P| + |S \cap P|$$. 16. Step 5: Substitute: $$90 + 75 + 45 - x + 5 = 145$$ $$210 - x + 5 = 145$$ $$215 - x = 145$$ $$x = 70$$ 17. Step 6: Number with exactly two amenities is: $$x - 3 \times |A \cap S \cap P| = 70 - 3 \times 5 = 70 - 15 = 55$$ 18. Final answer for problem 227 is 55. --- Final answers: - Problem 226: 8 integers less than 5 satisfy the inequality. - Problem 227: 55 houses have exactly two amenities.