Final answer:
To check the validity of the inequality x₂ + x₃ + 2x₄ ≤ 6 for the given constraint, we simplifying the inequality, solving it, substituting the range of values for x into the constraint equation, and determining the valid inequalities that satisfy the condition.
Step-by-step explanation:
Detailed Answer:
To determine if the inequality x₂ + x₃ + 2x₄ ≤ 6 is valid for X={(x ∈ ℤ₊⁴): 4x₁ + 5x₂ + 9x₃ + 12x₄ ≤ 34}, we need to find the range of values for x that satisfy both the inequality and the given constraint.
- First, simplify the inequality by completing the square in x², which reduces it to 2 (x² − ¹)² ≤ 0.
- Solving this inequality, we find that the maximum value for x² − 1 is 0, which means x² ≤ 1, resulting in -1 ≤ x ≤ 1.
- Next, substitute this range of values for x into the constraint equation 4x₁ + 5x₂ + 9x₃ + 12x₄ ≤ 34. By substituting the lower and upper bounds of x, we get -4 + 5x₂ + 9x₃ + 12x₄ ≤ 34 and 4 + 5x₂ + 9x₃ + 12x₄ ≤ 34, respectively.
- Solving these two inequalities, we find that -4 + 5x₂ + 9x₃ + 12x₄ ≤ 34 gives x₂ + x₃ + 2x₄ ≤ 6, while 4 + 5x₂ + 9x₃ + 12x₄ ≤ 34 does not satisfy the inequality.
Therefore, the inequality x₂ + x₃ + 2x₄ ≤ 6 is valid for X={(x ∈ ℤ₊⁴): 4x₁ + 5x₂ + 9x₃ + 12x₄ ≤ 34} when -1 ≤ x ≤ 1.