I'm supposed to solve the equation $$\frac{dy}{dx}=100-y$$ Now, by taking the right-hand expression to the left and the dx to the right, there's the obvious solution $$\ln{(100-y)}=-x+c_1$$ But I did this; $$\frac{dy}{dx}=-(y-100)$$$$\frac{dy}{y-100}=-dx$$$$\ln{(y-100)}=-x+c_2$$ Which is very much not the same. It's a first-order differential equation, and there should only be one root, so my method is obviously faulty.
Can someone help me with the why?
Hint: Note that $$\int \frac{1}{x}\mathrm dx = \ln\color{red}|x\color{red}| + C$$ Do not forget about the absolute value.