Question

Given startfraction (x minus 2) squared over 25 endfraction startfraction (y 3) squared over 4 endfraction less-than 1, which point lies in the solution set? (4, –0.5) (3, –2.5) (–2.5, 4) (–4.5, –3)

Answer 1

To determine which point lies in the solution set, substitute the values of x and y from each point into the inequality and check if the inequality is satisfied.

Step-by-step explanation:

To determine which point lies in the solution set of the given inequality, we need to substitute the values of x and y from each point into the inequality and check if the inequality is satisfied.

1. Substituting (4, -0.5):
2. Start by substituting x = 4 and y = -0.5 into the inequality:
3. 
   
4. ((4 - 2)² / 25) * ((-0.5 - 3)² / 4) < 1
5. 
   
6. Simplifying, we get:
7. 
   
8. (2² / 25) * (-3.5² / 4) < 1
9. 
   
10. (4/25) * (12.25/4) < 1
11. 
    
12. 1.225/25 < 1
13. 
    
14. 0.049 < 1
15. The inequality is true, so (4, -0.5) lies in the solution set.
16. Substituting (3, -2.5):
17. Start by substituting x = 3 and y = -2.5 into the inequality:
18. 
    
19. ((3 - 2)² / 25) * ((-2.5 - 3)² / 4) < 1
20. 
    
21. Simplifying, we get:
22. 
    
23. (1² / 25) * (-5.5² / 4) < 1
24. 
    
25. (1/25) * (30.25/4) < 1
26. 
    
27. 1.21/25 < 1
28. 
    
29. 0.0484 < 1
30. The inequality is true, so (3, -2.5) lies in the solution set.
31. Substituting (-2.5, 4):
32. Start by substituting x = -2.5 and y = 4 into the inequality:
33. 
    
34. ((-2.5 - 2)² / 25) * ((4 - 3)² / 4) < 1
35. 
    
36. Simplifying, we get:
37. 
    
38. (-4.5² / 25) * (1² / 4) < 1
39. 
    
40. (20.25/25) * (1/4) < 1
41. 
    
42. 0.81/25 < 1
43. 
    
44. 0.0324 < 1
45. The inequality is true, so (-2.5, 4) lies in the solution set.
46. Substituting (-4.5, -3):
47. Start by substituting x = -4.5 and y = -3 into the inequality:
48. 
    
49. ((-4.5 - 2)² / 25) * ((-3 - 3)² / 4) < 1
50. 
    
51. Simplifying, we get:
52. 
    
53. (-6.5² / 25) * (-6² / 4) < 1
54. 
    
55. (42.25/25) * (36/4) < 1
56. 
    
57. 1.69/25 < 1
58. 
    
59. 0.0676 < 1
60. The inequality is true, so (-4.5, -3) lies in the solution set.

Therefore, all four points (4, -0.5), (3, -2.5), (-2.5, 4), and (-4.5, -3) lie in the solution set of the given inequality.

Answer 2

The point (3, – 2.5) satisfies the given inequality and therefore lies in the solution set of the ellipse equation.

Step-by-step explanation:

The given equation resembles the standard form of an ellipse. To determine which point lies in the solution set, we have to plug the x and y values of each point into the equation ((x - 2)^2/25) + ((y + 3)^2/4) < 1 and see which one satisfies the inequality. Let's test each option:

For (4, – 0.5): ((4 - 2)^2/25) + ((– 0.5 + 3)^2/4) = (4/25) + (2.5^2/4) = 0.16 + 1.5625 = 1.7225 which is greater than 1, so it does not satisfy the inequality.

For (3, – 2.5): ((3 - 2)^2/25) + ((– 2.5 + 3)^2/4) = (1^2/25) + (0.5^2/4) = 0.04 + 0.0625 = 0.1025 which is less than 1, so this point is in the solution set.

For (–2.5, 4): ((-2.5 - 2)^2/25) + ((4 + 3)^2/4) = (20.25/25) + (49/4) = 0.81 + 12.25 = 13.06, which is greater than 1, so it does not satisfy the inequality.

For (–4.5, –3): ((-4.5 - 2)^2/25) + ((–3 + 3)^2/4) = (42.25/25) + (0^2/4) = 1.69 which is greater than 1, so it does not satisfy the inequality.

Therefore, the point that lies in the solution set is (3, – 2.5).