To solve the inequality cos(2x - π/6) ≥ -1/√2, we can first find the values of x for which cos(2x - π/6) = -1/√2.

Since cos(45°) = 1/√2, we can rewrite the inequality as cos(2x - π/6) ≥ cos(45°). This means that 2x - π/6 must be in the first or fourth quadrant where the cosine function is positive.

In the first quadrant, cosθ = cos(θ), so we have 2x - π/6 ≥ 45°. Solving for x, we get x ≥ (π/6 + 45°)/2 = (π/6 + π/4)/2 = (2π/12 + 3π/12)/2 = 5π/24. Similarly, in the fourth quadrant where cosθ = -cos(θ), we have 2x - π/6 ≤ -45°. Solving for x, we get x ≤ (π/6 - 45°)/2 = (π/6 - π/4)/2 = (2π/12 - 3π/12)/2 = -π/24.

Therefore, the solution to the inequality cos(2x - π/6) ≥ -1/√2 is -π/24 ≤ x ≤ 5π/24.