All points lying inside the triangle formed by the points (5, 0), (-1…

The correct answer is option (4) → (B), (D) and (E) Only

For a linear inequality to be satisfied by all points lying strictly inside a triangle, the inequality must hold true at all three vertices of the triangle.

Let the given vertices be:

• \(A = (5, 0)\)
• \(B = (-1, 2)\)
• \(C = (1, 3)\)

We test each linear inequality against these three vertices:

Statement (A): \(2x + 2y - 18 > 0\)

• At \(A(5, 0)\): \(2(5) + 2(0) - 18 = 10 - 18 = -8 ≯ 0\) (False)

Statement (B): \(3x + 2y > 0\)

• At \(A(5, 0)\): \(3(5) + 2(0) = 15 > 0\) (True)
• At \(B(-1, 2)\): \(3(-1) + 2(2) = 1 > 0\) (True)
• At \(C(1, 3)\): \(3(1) + 2(3) = 9 > 0\) (True)

Statement (C): \(2x + y + 13 < 0\)

• At \(A(5, 0)\): \(2(5) + 0 + 13 = 23 ≮0\) (False)

Statement (D): \(2x - 3y - 12 < 0\)

• At \(A(5, 0)\): \(2(5) - 3(0) - 12 = -2 < 0\) (True)
• At \(B(-1, 2)\): \(2(-1) - 3(2) - 12 = -20 < 0\) (True)
• At \(C(1, 3)\): \(2(1) - 3(3) - 12 = -19 < 0\) (True)

Statement (E): \(2x - 3y + 12 > 0\)

• At \(A(5, 0)\): \(2(5) - 3(0) + 12 = 22 > 0\) (True)
• At \(B(-1, 2)\): \(2(-1) - 3(2) + 12 = 4 > 0\) (True)
• At \(C(1, 3)\): \(2(1) - 3(3) + 12 = 5 > 0\) (True)

Conclusion

Statements (B), (D), and (E) evaluate to true for all three corner vertices, ensuring the entire interior space of the triangle fulfills these constraints.