To solve the first inequality:

3x + 4 ≤ -x + 12
Add x to both sides:
3x + x + 4 ≤ 12
4x + 4 ≤ 12
Subtract 4 from both sides:
4x ≤ 8
Divide by 4:
x ≤ 2

So the solution for the first inequality is x ≤ 2.

To solve the second inequality:

5x - 7 < 7x + 3
Subtract 5x from both sides:
-7 < 2x + 3
Subtract 3 from both sides:
-10 < 2x
Divide by 2:
-5 < x

So the solution for the second inequality is x > -5.

Therefore, the solutions to the system of inequalities are -5 < x ≤ 2.