Why would the derivate of u - 2 = x be du = xdx?

Question

I am going through an integration requiring a u-substitution on a practice midterm, and the professor posted a solution which used:

u = x - 2

u - 2 = x

du = xdx

I am confused how du could be anything other than dx? From what I understand, the derivative of u - 2 and the derivative of u are identical. The way he did this was key to how he solved the problem. Here's the solution:

Answer 1

Note that $$u=x+2 \implies \frac d{dx} (u)=\frac d{dx}(x+2) \implies \frac{du}{dx}=1\implies du=dx$$

What was written above in your book's solution ($du=x\,dx$) may be a typo.