Why is the characteristic of (d/dx) + (d/dt) = 0 not c = x + t?

[Snippy]

Why is the characteristic of

(d/dx) + (d/dt) = 0 where d is small delta

c = x - t and not c = x + t

 

[elibj123]

Because$$(\partial_{x}+\partial_{t})f(x+t)=2f'(x+t)$$

While

$$(\partial_{x}+\partial_{t})f(x-t)=f'(x-t)-f'(x-t)=0$$

So you need solutions of the latter form u(x,t)=f(x-t), which means x-t=const are charcteristic lines.