Maximizing Production of Widgets

A factory produces two types of widgets, Type X and Type Y. Let X represent the number of Type X widgets produced and Y denote the number of Type Y widgets produced. The factory has established the following production constraints:

- The total number of widgets produced cannot exceed 150.

- The production of Type Y widgets must be at least twice that of Type X widgets, i.e., $Y \geq 2X$.

- The production cost is $15 for each Type X widget and $10 for each Type Y widget. The factory has a budget of $1,200 for production costs.

Given these constraints, what is the maximum number of Type X widgets that the factory can produce?