To find the set of values of \(k\) for which the line does not meet the curve, we need to solve the system of equations:

1. \(y = 2x + \frac{12}{x}\)

2. \(y + x = k\)

Substitute \(y = k - x\) from the second equation into the first equation:

\(k - x = 2x + \frac{12}{x}\)

Rearrange to form a quadratic equation:

\(2x + \frac{12}{x} = k - x\)

\(2x^2 + 12 = kx - x^2\)

\(3x^2 - kx + 12 = 0\)

For the line not to meet the curve, the quadratic equation must have no real solutions. This occurs when the discriminant \(b^2 - 4ac\) is less than zero.

Here, \(a = 3\), \(b = -k\), \(c = 12\).

The discriminant is:

\((-k)^2 - 4 \times 3 \times 12\)

\(k^2 - 144 \lt 0\)

Solving \(k^2 - 144 \lt 0\) gives:

\(-12 \lt k \lt 12\)