To solve these inequalities, we need to isolate the variable x on one side of the inequality sign.

1) 25 - 6x ≤ 4 + x

First, let's simplify the inequality:

25 - 6x ≤ 4 + x
Subtract 25 from both sides:
-6x ≤ -21 + x
Combine like terms:
-6x - x ≤ -21
-7x ≤ -21
Divide both sides by -7 (remember to flip the inequality sign when dividing by a negative number):
x ≥ 3

Therefore, the solution to the first inequality is x ≥ 3.

2) 3x + 7 < 1 + 4x

Let's simplify the inequality:

3x + 7 < 1 + 4x
Subtract 3x from both sides:
7 < 1 + x
Subtract 1 from both sides:
6 < x

Therefore, the solution to the second inequality is x > 6.

To summarize:
For the inequality 25 - 6x ≤ 4 + x, the solution is x ≥ 3.
For the inequality 3x + 7 < 1 + 4x, the solution is x > 6.