To solve the equation cos(x-pi/6) = -sqrt(2)/2, we can use the inverse cosine function (also known as arccosine) to find the value of x.

First, let's find the reference angle for -sqrt(2)/2. The reference angle is the angle between the terminal side of the angle and the x-axis in the standard position.

The cosine function is negative in the second and third quadrants. In the second quadrant, the reference angle for -sqrt(2)/2 is pi/4.

Now, let's find the general solution for x by adding or subtracting multiples of 2pi to the reference angle:

x = pi/4 + 2pi*n, where n is an integer.

However, we need to consider the shift of pi/6 in the equation cos(x-pi/6) = -sqrt(2)/2.

To find the specific solution, we substitute the reference angle into the equation:

x - pi/6 = pi/4 + 2pi*n

Solving for x:

x = pi/4 + pi/6 + 2pi*n

x = (3pi/12 + 2pi/12) + 2pi*n

x = 5pi/12 + 2pi*n

Therefore, the specific solution for the equation cos(x-pi/6) = -sqrt(2)/2 is:

x = 5pi/12 + 2pi*n, where n is an integer.