Question

A company sells gadgets for $20/gadget and has a profit function from making and selling x gadgets of P(x) = 15x - 6000.

Answer 1

This is the concept of algebra, at breakeven point there are no profits made. Hence, given the profit function 
P(x)=15x-6000
at breakeven point the number of units sold will be:
P(x)=0
thus:
0=15x-6000
15x=6000
x=400
the number of units sold at breakeven will be 400 units.
 The amount at breakeven wil be:
number of units x price of each unit
=400x20
=$ 8000

The results found are true breakeven point because at these points the company will not be generating any profits because the amount of revenue generated will be equal to the amount of expenses. This means that the difference between revenue and cost will be zero, therefor it satisfies the breakeven point rationale.