Gadget Production Constraint Optimization

A company manufactures two types of gadgets: Type X and Type Y. The cost to produce one Type X gadget is $15, and the cost to produce one Type Y gadget is $25. The company can allocate a total budget of $600 for production.

If the company wants to produce at least 10 gadgets in total but no more than 30 gadgets, let $x$ represent the number of Type X gadgets produced, and $y$ represent the number of Type Y gadgets produced.

Formulate the inequality representing the constraints on production and determine the maximum number of Type Y gadgets that can be produced when the total number of gadgets is exactly 30.