# How to find the least number of objects in a problem involving profits?

The problem is as follows:

In an electronics factory, the owner calculates that the cost to produce his new model of portable TV is $26$ dollars. After meeting with the distributors, he agrees the sale price for his new product to be $25$ dollars each and additionally $8\%$ more for each TV set sold after $8000$ units. What is the least number of TV's he has to sell in order to make a profit?.

• 16000
• 15001
• 16001
• 15999
• 17121

This problem has made me to go in circles on how to express it in a mathematical expression. I'm not sure if it does need to use of inequations.

What I tried to far is to think this way:

The first scenario is what if what he sells is $8000$ units, then this would become into:

$$\textrm{production cost:}\,26\frac{\}{\textrm{unit}} \times 8000\,\textrm{units} = 208000\,\$$$$$\textrm{sales:}\,25\frac{\}{\textrm{unit}} \times 8000\,\textrm{units}=\,200000\,\$$$

Therefore there will be an offset of $8000\,\$$as$$208000\$-200000\$\,=\,8000\,\$$So I thought what If I consider the second part of the problem which it says that he will receive an additional of 8\% after 8000 units. Therefore his new sale price will be 27\,\$$ because: $$25+\frac{8}{100}\left(25\right )=27\,\$$$

So from this I thought that this can be used in the previous two relations. But how?.

I tried to establish this inequation:

$$26\left(8000+x\right)<25\left(8000\right)+27\left(8000+x\right)$$

But that's where I'm stuck at since it is not possible to obtain a reasonable result from this as one side will be negative and the other positive.