А) |4x - x^2| > 4 - x

To solve this inequality, first factor out a negative sign from the absolute value:

|x(4 - x)| > 4 - x

Now consider two cases:

Case 1: x < 4
In this case, the absolute value can be written as x(4 - x):

x(4 - x) > 4 - x
4x - x^2 > 4 - x
x^2 - 5x + 4 < 0
(x - 4)(x - 1) < 0
1 < x < 4

Case 2: x > 4
In this case, the absolute value can be written as x(x - 4):

x(x - 4) > 4 - x
x^2 - 4x > 4 - x
x^2 - 3x - 4 > 0
(x - 4)(x + 1) > 0
x < -1 or x > 4

Therefore, the solution to the inequality |4x - x^2| > 4 - x is x < -1 or 1 < x < 4.

B) |x^2 - 9| ≤ 6 - 2x

To solve this inequality, first factor out the absolute value:

|x^2 - 9| ≤ 6 - 2x

Now consider two cases:

Case 1: x ≤ 3
In this case, the absolute value can be written as |x - 3||x + 3|:

(x - 3)(x + 3) ≤ 6 - 2x
x^2 - 9 ≤ 6 - 2x
x^2 + 2x - 15 ≤ 0
(x + 5)(x - 3) ≤ 0
-5 ≤ x ≤ 3

Case 2: x > 3
In this case, the absolute value can be written as |x + 3|:

-(x + 3) ≤ 6 - 2x
-x - 3 ≤ 6 - 2x
x ≤ 9

Therefore, the solution to the inequality |x^2 - 9| ≤ 6 - 2x is -5 ≤ x ≤ 3 or x ≤ 9.