A number is doubled, then decreased by $13.7$.  The result is greater than $125.28$. What is the smallest integer that satisfies this condition?
Let $x$ be the desired integer. Then $2x-13.7>125.28$.  Adding $13.7$ to both sides gives $2x>138.98$, and dividing both sides by $2$ gives $x>69.49$.  The smallest integer that is greater than $69.49$ is $\boxed{70}$.