Final answer:
The solution set of the quadratic inequality 2z^2 - 2z + 12 ≤ 36 is -3 ≤ z ≤ 4, after factoring and testing intervals.
Step-by-step explanation:
Let us start by correcting the inequality which appears to have a typo. Assuming it should be 2z^2 - 2z + 12 ≤ 36, we can solve the quadratic inequality step-by-step.
- Subtract 36 from both sides to get the quadratic expression on one side: 2z^2 - 2z - 24 ≤ 0.
- Factor the quadratic equation: (2z + 6)(z - 4) ≤ 0.
- Determine the critical values by setting each factor to zero: 2z + 6 = 0 gives z = -3, and z - 4 = 0 gives z = 4.
- Test intervals to find where the inequality holds true. Select test points from intervals (−∞, -3), (−3, 4), and (4, ∞).
- Using a sign chart or graphing calculator, we can determine that the inequality holds true for -3 ≤ z ≤ 4.
Therefore, the main answer is that the solution set of the inequality is -3 ≤ z ≤ 4.