To solve this logarithmic inequality, we can rewrite it in exponential form:

0.5^(Log0.5(6|x| - 3)) ≤ 0.5^(Log0.5(4-x^2))

Since the bases are the same, we can simplify this to:

6|x| - 3 ≤ 4 - x^2

Now we can solve for x by isolating it on one side:

6|x| + x^2 - 3 - 4 ≤ 0

6|x| + x^2 - 7 ≤ 0

Now we need to consider two cases:

Case 1: x is positive or zero

6x + x^2 - 7 ≤ 0
x^2 + 6x - 7 ≤ 0
(x+7)(x-1) ≤ 0

This inequality is true for -7 ≤ x ≤ 1.

Case 2: x is negative

6(-x) + x^2 - 7 ≤ 0
x^2 - 6x - 7 ≤ 0
(x+1)(x-7) ≤ 0

This inequality is true for -7 ≤ x ≤ 1.

Therefore, the solution to the original inequality is -7 ≤ x ≤ 1.