We can solve the inequality as follows:
-2(3x+6) ≥ 4(x+7)
-6x-12 ≥ 4x+28 (distribute -2 on the left and simplify on the right)
-10x-12 ≥ 28 (combine like terms)
-10x ≥ 40 (add 12 to both sides)
x ≤ -4 (divide by -10 and flip the inequality sign)
Therefore, the inequality that represents all the solutions of the given inequality is x ≤ -4.