The first step is to simplify the expression on the right side of the inequality:
4(3x-18)-9 = 12x - 81
Substituting this into the original inequality gives:
a(2x-12) < 12x - 81
Now we can simplify the left side of the inequality:
a(2x-12) = 2ax - 12a
The inequality now reads:
2ax - 12a < 12x - 81
Simplifying, we get:
2ax - 12x < 81 - 12a
Now we can solve for a:
2x(a - 6) < 81 - 12a
2x(a - 6) + 12a - 81 < 0
2ax - 6x - 81 < 0
2x(a - 3) - 3(a - 3) - 81 < 0
(2x - 3)(a - 3) < 81
For this inequality to hold for all real values of x, the left side must be negative, zero, or positive for all x. Since 2x - 3 can take on any real value, we need a - 3 to be such that the product is always less than 81. Thus:
a - 3 < 0
a < 3
Therefore, the value of a that ensures that the inequality has a solution of all real numbers is a < 3.