Final answer:
The maximum possible integer value of k for the equation with no real solutions is -442, considering that the discriminant must be negative with j being greater than k.
Step-by-step explanation:
To find the maximum possible integer value of k when given an equation with no real solutions, we need to look at the discriminant of the quadratic equation.
The equation provided is -x² + 42x + j = 0. For a quadratic equation ax² + bx + c = 0, the discriminant is given by b² - 4ac.
For the equation to have no real solutions, the discriminant must be negative, so we should have 42² - 4(-1)j < 0.
After calculating, we get 1764 + 4j < 0. If we solve for j, we find that j < -441. Since we know that j > k, the maximum possible integer value for k would be just less than -441, which is -442.