### Video Transcript

A phone manufacturer has a cost of production function πΆ of π₯ equal to 11π₯ plus 120 and a revenue function π
of π₯ equal to five π₯, where π₯ is the number of phones it makes, and πΆ of π₯ and π
of π₯ are in dollars. How many phones does the company need to manufacture in order to make a profit?

The break-even point for any company occurs when the cost is equal to the revenue. This means that in order to make a profit, the revenue, π
of π₯, must be greater than the cost, πΆ of π₯. In this question, the revenue or π
of π₯ is equal to five π₯. The cost of production is 11π₯ plus 120. In order to make a profit, five π₯ needs to be greater than 11π₯ plus 120.

We can begin to solve this inequality by subtracting five π₯ from both sides. This gives us zero is greater than six π₯ plus 120 or six π₯ plus 120 is less than zero. We can then subtract 120 from both sides, giving us six π₯ is less than negative 120. Finally, we divide both sides of the inequality by six. Dividing a negative number by a positive number gives a negative answer. Therefore, π₯ is less than negative 20. π₯ is the number of phones that the manufacturer makes. We know that this must be positive. Therefore, π₯ cannot be less than negative 20.

We can, therefore, conclude that this company will never be able to make a profit with these current costs of production. This is because the cost, πΆ of π₯, will always be greater than the revenue, π
of π₯. It doesnβt matter how many phones the company manufactures. They will never make a profit.