1. Solve the equation: $$6 - \left[\frac{3}{2} \div \left(0.5 + \frac{1}{4}\right)\right] = 4$$ 2. First, simplify inside the parentheses: $$0.5 + \frac{1}{4} = 0.5 + 0.25 = 0.75$$ 3. Now divide $$\frac{3}{2}$$ by 0.75: $$\frac{3}{2} \div 0.75 = \frac{3}{2} \times \frac{1}{0.75} = \frac{3}{2} \times \frac{4}{3} = \cancel{\frac{3}{\cancel{2}}} \times \frac{4}{\cancel{3}} = 2$$ 4. Substitute back into the equation: $$6 - 2 = 4$$ 5. Check if the equation holds: $$4 = 4$$ which is true. --- 6. Solve: $$4.8 \div \frac{3}{5}$$ 7. Convert division by a fraction to multiplication by its reciprocal: $$4.8 \div \frac{3}{5} = 4.8 \times \frac{5}{3}$$ 8. Calculate: $$4.8 \times \frac{5}{3} = \frac{4.8 \times 5}{3} = \frac{24}{3} = 8$$ --- 9. Simplify: $$(3x^2 - 5x + 7) + (2x^2 + x - 4)$$ 10. Combine like terms: - For $$x^2$$ terms: $$3x^2 + 2x^2 = 5x^2$$ - For $$x$$ terms: $$-5x + x = -4x$$ - For constants: $$7 - 4 = 3$$ 11. Final simplified expression: $$5x^2 - 4x + 3$$ --- 12. Simplify: $$-2(x^2 - 4x + 1) + 3(x^2 + x - 6)$$ 13. Distribute: $$-2x^2 + 8x - 2 + 3x^2 + 3x - 18$$ 14. Combine like terms: - $$-2x^2 + 3x^2 = x^2$$ - $$8x + 3x = 11x$$ - $$-2 - 18 = -20$$ 15. Final expression: $$x^2 + 11x - 20$$ --- 16. Simplify: $$(4x)(3x^2 - 5x + 2)$$ 17. Distribute $$4x$$: $$4x \times 3x^2 = 12x^3$$ $$4x \times (-5x) = -20x^2$$ $$4x \times 2 = 8x$$ 18. Final expression: $$12x^3 - 20x^2 + 8x$$ --- 19. Simplify: $$(3x^2 - x) + (2x - 5) - (x^2 + 4)$$ 20. Remove parentheses and change signs for the last group: $$3x^2 - x + 2x - 5 - x^2 - 4$$ 21. Combine like terms: - $$3x^2 - x^2 = 2x^2$$ - $$-x + 2x = x$$ - $$-5 - 4 = -9$$ 22. Final expression: $$2x^2 + x - 9$$ --- 23. Arrange the rational numbers $$0.401, \frac{2}{5}, 0.41, 0.411$$ from least to greatest. 24. Convert $$\frac{2}{5}$$ to decimal: $$\frac{2}{5} = 0.4$$ 25. Order the decimals: $$0.4 < 0.401 < 0.41 < 0.411$$ 26. Final order: $$\frac{2}{5}, 0.401, 0.41, 0.411$$ --- 27. Find the height $$h$$ of a triangle with area $$45x^2y^2$$ and base $$10x$$. 28. Use the area formula for a triangle: $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$ 29. Substitute known values: $$45x^2y^2 = \frac{1}{2} \times 10x \times h$$ 30. Simplify right side: $$45x^2y^2 = 5x h$$ 31. Solve for $$h$$: $$h = \frac{45x^2y^2}{5x} = 9xy^2$$ --- 32. Express $$\frac{1}{75}$$ in lowest terms. 33. Since 1 and 75 have no common factors other than 1, the fraction is already in lowest terms: $$\frac{1}{75}$$ --- 34. Simplify: $$1 \frac{1}{2} + \frac{3}{4} + \frac{3}{4} + 2.5 \times \left(\frac{2}{5} - 0.1\right)$$ 35. Convert mixed number to improper fraction: $$1 \frac{1}{2} = \frac{3}{2} = 1.5$$ 36. Calculate inside parentheses: $$\frac{2}{5} - 0.1 = 0.4 - 0.1 = 0.3$$ 37. Multiply: $$2.5 \times 0.3 = 0.75$$ 38. Sum all terms: $$1.5 + \frac{3}{4} + \frac{3}{4} + 0.75 = 1.5 + 0.75 + 0.75 + 0.75 = 3.75$$ --- 39. Simplify: $$(6a - 4ax^2 + 9) - (2x^2 - 3a + 5)$$ 40. Remove parentheses and change signs: $$6a - 4ax^2 + 9 - 2x^2 + 3a - 5$$ 41. Combine like terms: - $$6a + 3a = 9a$$ - $$-4ax^2$$ (no like term) - $$9 - 5 = 4$$ - $$-2x^2$$ (no like term) 42. Final expression: $$9a - 4ax^2 - 2x^2 + 4$$ --- 43. Simplify: $$(2x^5 x^3)(x^2 - x - 5)$$ 44. Multiply powers of $$x$$: $$2x^{5+3} = 2x^8$$ 45. Distribute: $$2x^8 \times x^2 = 2x^{10}$$ $$2x^8 \times (-x) = -2x^9$$ $$2x^8 \times (-5) = -10x^8$$ 46. Final expression: $$2x^{10} - 2x^9 - 10x^8$$ --- 47. Simplify: $$\frac{36x^5 + 12x^3}{6^B} = 6x^2 + 2$$ 48. Factor numerator: $$12x^3(3x^2 + 1)$$ 49. Write equation: $$\frac{12x^3(3x^2 + 1)}{6^B} = 6x^2 + 2$$ 50. To find $$B$$, equate powers and coefficients. Since $$6^B$$ divides numerator to get right side, and right side is linear in $$x^2$$, the value of $$B$$ is 2. --- 51. Simplify expression for perimeter of rectangle with length $$3x + 7$$ and width $$x - 2$$. 52. Perimeter formula: $$P = 2(\text{length} + \text{width})$$ 53. Substitute: $$P = 2((3x + 7) + (x - 2)) = 2(4x + 5)$$ 54. Distribute: $$P = 8x + 10$$ --- 55. Write expression for area model (not fully specified in prompt, so no further simplification).