You need to find the equations for f(x+1),g(x+2),g(x+7).
Finding f(x+1) means to plug x+1 in equation of f(x) such that:
Expanding yields: f(x+1)=2x+2-4 => f(x+1)=2x-2
Finding g(x+2) means to plug x+2 in equation of g(x) such that:
g(x+2)=-(x+2)+5 => g(x+2)=-x-2+5 => g(x+2)=-x+3
Finding g(x+7) means to plug x+7 in equation of g(x) such that:
g(x+7)=-(x+7)+5 => g(x+7)=-x-7+5 => g(x+7)=-x-2
Plugging equations of f(x+1),g(x+2),g(x+7) in 3f(x+1)-2g(x+2)=f(x)-g(x+7) yields:
3(2x-2) - 2(-x+3) = 2x - 4 - (-x - 2)
6x - 6 + 2x - 6 = 2x - 4 + x + 2
Reducing 2x both sides yields:
6x - 12 = x - 2
You need to move the terms containing x to the left side such that:
6x - x = 12 - 2 => 5x = 10 => x = 2
Hence, the solution to the equation 3f(x+1)-2g(x+2)=f(x)-g(x+7) is x = 2.