To solve this equation, we need to consider the different cases based on the signs of x+7 and x-4.

Case 1: x+7 ≥ 0 and x-4 ≥ 0
In this case, both absolute values are positive, so we can remove the absolute value signs:
(x+7) - (x-4) = 9
x + 7 - x + 4 = 9
11 = 9
This is not possible, so there are no solutions in this case.

Case 2: x+7 ≥ 0 and x-4 < 0
In this case, the absolute value of x+7 is positive, but the absolute value of x-4 is negative, so we need to change the sign of the second absolute value:
(x+7) + (x-4) = 9
2x + 3 = 9
2x = 6
x = 3

Case 3: x+7 < 0 and x-4 ≥ 0
In this case, the absolute value of x+7 is negative, but the absolute value of x-4 is positive, so we need to change the sign of the first absolute value:
-(x+7) - (x-4) = 9
-2x - 11 = 9
-2x = 20
x = -10

Case 4: x+7 < 0 and x-4 < 0
In this case, both absolute values are negative, so we can remove the absolute value signs:
-(x+7) + (x-4) = 9
-2x + 3 = 9
-2x = 6
x = -3

Therefore, the solutions to the equation |x+7|-|x-4|=9 are x = 3, -10, and -3.