1) !x-3!=4;2) !x-4!=-3;3) !x+4!=3;4) !x+3!=4;

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To solve these factorial equations, we must first understand the properties of factorials.

The factorial of a number n, denoted as n!, is the product of all positive integers from 1 to n. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.

Let's solve each of the equations:

1) !x-3!=4
Let's first express 4 as 4! = 4 x 3 x 2 x 1 = 24
So, !x - 3! = 24
!x - 6 = 24
!x = 30
Therefore, x = 4 because 4! = 24.

2) !x-4!=-3
Let's first simplify -3 as -3! = -3 x 2 x 1 = -6
So, !x - 4! = -6
!x - 24 = -6
!x = 18
Therefore, x = 3 because 3! = 6.

3) !x+4!=3
Let's express 3 as 3! = 3 x 2 x 1 = 6
So, !x + 4! = 6
!x + 24 = 6
!x = -18
This equation does not have a real solution since factorials are not defined for negative values.

4) !x+3!=4
Let's express 4 as 4! = 4 x 3 x 2 x 1 = 24
!x + 6 = 24
!x = 18
Therefore, x = 3 because 3! = 6.

I hope this helps! Let me know if you have any more questions.