To solve this inequality, I need to isolate $x$ on one side of the inequality sign. I can start by adding $7$ to both sides, which gives me $x+7<2x$. Then I can subtract $x$ from both sides, which gives me $7<x$. This means that $x$ must be greater than $7$, but I want the smallest such integer. The smallest integer that is greater than $7$ is $8$, so $x=8$ is the smallest solution. 
What is the smallest integer $x$ for which $x<2x-7$ ?