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 ≤ 0x^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 ≤ 0x^2 - 6x - 7 ≤ 0(x+1)(x-7) ≤ 0
Therefore, the solution to the original inequality is -7 ≤ x ≤ 1.
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.