1) To solve x + 4 = |x - 7|, we need to consider two cases:

Case 1: When x - 7 is greater than or equal to 0 (x - 7 >= 0)
x + 4 = x - 7
x + 4 - x = -7
4 = -7 (Not possible)

Case 2: When x - 7 is less than 0 (x - 7 < 0)
x + 4 = -(x - 7)
x + 4 = -x + 7
x + x = 7 - 4
2x = 3
x = 3/2

Therefore, the solution to x + 4 = |x - 7| is x = 3/2.

2) To solve x + 1 = |x - 9|, we need to consider two cases as well:

Case 1: When x - 9 is greater than or equal to 0 (x - 9 >= 0)
x + 1 = x - 9
x + 1 - x = -9
1 = -9 (Not possible)

Case 2: When x - 9 is less than 0 (x - 9 < 0)
x + 1 = -(x - 9)
x + 1 = -x + 9
x + x = 9 - 1
2x = 8
x = 4

Therefore, the solution to x + 1 = |x - 9| is x = 4.