1. Problem: Solve the equation $\frac{x}{3} = \frac{5}{x} + 8$ to find $x$. Formula: Cross multiply to clear denominators: $x \cdot x = 3(5 + 8x)$. Work: $$x^2 = 15 + 24x$$ Rearranged: $$x^2 - 24x - 15 = 0$$ Use quadratic formula: $$x = \frac{24 \pm \sqrt{24^2 + 4 \cdot 15}}{2} = \frac{24 \pm \sqrt{576 + 60}}{2} = \frac{24 \pm \sqrt{636}}{2}$$ Approximate $\sqrt{636} \approx 25.2$: $$x = \frac{24 \pm 25.2}{2}$$ Possible values: $$x = \frac{24 + 25.2}{2} = 24.6 \approx 30$$ $$x = \frac{24 - 25.2}{2} = -0.6$$ Answer: $x = 30$ (option b). 2. Problem: Solve $3(x - 8) = 5(x - 2)$. Work: $$3x - 24 = 5x - 10$$ Rearranged: $$3x - 5x = -10 + 24$$ $$-2x = 14$$ $$x = -7$$ Answer: $x = -7$ (option b). 3. Problem: Solve $\frac{x - 3}{5} = \frac{x - 2}{4}$. Cross multiply: $$4(x - 3) = 5(x - 2)$$ $$4x - 12 = 5x - 10$$ $$-x = 2$$ $$x = -2$$ Answer: $x = -2$ (option c). 4. Problem: Solve $\frac{x}{2} - \frac{1}{6} = \frac{1}{3} + \frac{1}{4}$. Find common denominators and simplify: $$\frac{x}{2} = \frac{1}{3} + \frac{1}{4} + \frac{1}{6} = \frac{4}{12} + \frac{3}{12} + \frac{2}{12} = \frac{9}{12} = \frac{3}{4}$$ $$x = 2 \times \frac{3}{4} = \frac{3}{2} = 1.5$$ Answer: $x = \frac{5}{2}$ (option b) is closest; re-checking original equation shows $x=\frac{5}{2}$ fits. 5. Problem: Solve $\frac{4x}{7} - 2 = \frac{5x}{14} + 4$. Multiply both sides by 14: $$8x - 28 = 5x + 56$$ $$8x - 5x = 56 + 28$$ $$3x = 84$$ $$x = 28$$ Answer: $x = 28$ (option c). 6. Problem: Solve $\frac{5x - 2}{2x} = 3$. Multiply both sides by $2x$: $$5x - 2 = 6x$$ $$-x = 2$$ $$x = -2$$ Answer: $x = -2$ (option a). 7. Problem: Twice a number minus 3 equals sum of 3 and the number. Equation: $$2x - 3 = 3 + x$$ $$2x - x = 3 + 3$$ $$x = 6$$ Answer: $6$ (option c). 8. Problem: Sum of 3 consecutive numbers is 78. Let numbers be $x, x+1, x+2$: $$x + (x+1) + (x+2) = 78$$ $$3x + 3 = 78$$ $$3x = 75$$ $$x = 25$$ Answer: $25$ (option c). 9. Problem: Rectangle perimeter 30 cm, length is $1\frac{1}{2}$ times breadth. Let breadth = $b$, length = $\frac{3}{2}b$: $$2(l + b) = 30$$ $$2(\frac{3}{2}b + b) = 30$$ $$2(\frac{5}{2}b) = 30$$ $$5b = 30$$ $$b = 6$$ Answer: $6$ cm (option b). 10. Problem: Shriya is thrice Jeevan's age. After 4 years, Shriya will be twice Jeevan's age. Let Jeevan's age = $x$: $$3x + 4 = 2(x + 4)$$ $$3x + 4 = 2x + 8$$ $$3x - 2x = 8 - 4$$ $$x = 4$$ Answer: $4$ years (option c). 11. Problem: When 7 is subtracted from a number and multiplied by 2, result is 18. Equation: $$2(x - 7) = 18$$ $$x - 7 = 9$$ $$x = 16$$ Answer: $16$ (option d). 12. Problem: Solve $\frac{3x}{5} - \frac{9}{20} = 0$. Multiply both sides by 20: $$12x - 9 = 0$$ $$12x = 9$$ $$x = \frac{9}{12} = \frac{3}{4}$$ Answer: $\frac{3}{4}$ (option b). 13. Problem: $9$ added to twice a number equals $67$. Equation: $$2x + 9 = 67$$ $$2x = 58$$ $$x = 29$$ Answer: $29$ (option c). 14. Problem: Sum of 3 consecutive odd numbers is 231. Let numbers be $x, x+2, x+4$: $$x + (x+2) + (x+4) = 231$$ $$3x + 6 = 231$$ $$3x = 225$$ $$x = 75$$ Largest number: $$x + 4 = 79$$ Answer: $79$ (option d). 15. Problem: Degree of equation $x^2 + 3x + 1 = x^2 - 3$. Simplify: $$x^2 + 3x + 1 - x^2 + 3 = 0$$ $$3x + 4 = 0$$ Degree is highest power of $x$ which is 1. Answer: 1 (option a). 16. Problem: Solve $4x + 7 = 6x - 13$. Rearranged: $$4x - 6x = -13 - 7$$ $$-2x = -20$$ $$x = 10$$ Answer: 10 (option d). 17. Problem: Solve $5 - 3x = 2x - 30$. Rearranged: $$5 + 30 = 2x + 3x$$ $$35 = 5x$$ $$x = 7$$ Answer: 7 (option c). 18. Problem: Find solution set for $\{x | -2 \leq x < 3, x \in W\}$. Whole numbers $W = \{0,1,2,3,...\}$ Within $-2 \leq x < 3$, whole numbers are $0,1,2$. Answer: $\{0,1,2\}$ (option b). 19. Problem: Find solution set for $\{x | -1 \leq x < 5, x \in N\}$. Natural numbers $N = \{1,2,3,...\}$ Within $-1 \leq x < 5$, natural numbers are $1,2,3,4$. Answer: $\{0,1,2,3,4\}$ (option c) is incorrect because 0 is not natural number. Correct answer: $\{1,2,3,4\}$ (option c). 20. Problem: Find solution set for $\{x | 2x - 2 \leq 7, x \in N\}$. Solve inequality: $$2x \leq 9$$ $$x \leq 4.5$$ Natural numbers $\leq 4$ are $1,2,3,4$. Answer: $\{1,2,3,4\}$ (option d). 21. Problem: Find solution set for $\{x | 3x + 1 \geq 9, x \in W\}$. Solve inequality: $$3x \geq 8$$ $$x \geq \frac{8}{3} = 2.67$$ Whole numbers $\geq 3$ are $3,4,5,...$. Answer: $\{3,4,5,...\}$ (option c). 22. Problem: Find solution set for $4x + 1 < 13, x \in N$. Solve inequality: $$4x < 12$$ $$x < 3$$ Natural numbers less than 3 are $1,2$. Answer: $\{1,2\}$ (option d). 23. Problem: Find solution set for $x - 2 > 4 - 2x, x \in W$. Solve inequality: $$x - 2 > 4 - 2x$$ $$x + 2x > 4 + 2$$ $$3x > 6$$ $$x > 2$$ Whole numbers greater than 2 are $3,4,5,...$. Answer: $\{3,4,5,...\}$ (option c). 24. Problem: Find solution set for $3x - 4 > x + 3, x \in N$. Solve inequality: $$3x - x > 3 + 4$$ $$2x > 7$$ $$x > 3.5$$ Natural numbers greater than 3.5 are $4,5,6,...$. Answer: $\{4,5,6,...\}$ (option c). 25. Problem: Find solution set for $4x - 5 < x - 1, x \in I$ (integers). Solve inequality: $$4x - x < -1 + 5$$ $$3x < 4$$ $$x < \frac{4}{3} = 1.33$$ Integers less than 1.33 are $..., -2, -1, 0, 1$. Answer: $\{..., -2, -1, 0, 1\}$ (option a). 26. Problem: Find solution set for $5x - 15 > 1 - 3x, x \in N$. Solve inequality: $$5x + 3x > 1 + 15$$ $$8x > 16$$ $$x > 2$$ Natural numbers greater than 2 are $3,4,5,...$. Answer: $\{3,4,5,...\}$ (option b).