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Bot GPT 3.5 To check which point is a solution to the simultaneous inequalities, we need to substitute the x and y values of each point into both inequalities and see if both inequalities are true.
a. (25,12):
For y < 1/2x: 12 < 1/2(25) → 12 < 12.5 (true)
For y > 1/3x + 3: 12 > 1/3(25) + 3 → 12 > 8.33 + 3 → 12 > 11.33 (true)
Both inequalities are true, so (25,12) is a solution.
b. (2,2):
For y < 1/2x: 2 < 1/2(2) → 2 < 1 (false)
For y > 1/3x + 3: 2 > 1/3(2) + 3 → 2 > 0.67 + 3 → 2 > 3.67 (false)
Both inequalities are false, so (2,2) is not a solution.
c. (10,5):
For y < 1/2x: 5 < 1/2(10) → 5 < 5 (false)
For y > 1/3x + 3: 5 > 1/3(10) + 3 → 5 > 3.33 + 3 → 5 > 6.33 (false)
Both inequalities are false, so (10,5) is not a solution.
d. (3,6):
For y < 1/2x: 6 < 1/2(3) → 6 < 1.5 (false)
For y > 1/3x + 3: 6 > 1/3(3) + 3 → 6 > 1 + 3 → 6 > 4 (true)
The first inequality is false while the second inequality is true, so (3,6) is not a solution.
The only point that is a solution to both inequalities is (25,12), so the answer is (a).