Algebra....Break even point?

Question

I'm helping my daughter with a math problem......here's the question:

Expenses = 5 + 12x
Income = 25 + 14x
Therefore, Profit = (25 +14x) - (5 +12x) right?

Now the question is.....How many sweatshirts "x" must be sold to "break even"?

The way I worked the problem was to make Profit "P" equal to zero. But when I do this, I come out with x = -10 shirts. I know that can't be right. What are we doing wrong? Please explain in detail so we can understand where we went wrong. Thanks!

Answer 1

You are correct. The answer is that even if you sell 0 shirts, you will still make a profit based on this equation.

Take into account that you sell zero shirts. Your income would be 25 and your expense would be 5. The equations provided can not be a realistic view of a business.

The guy above me has not done his math correctly. He indicated that 0=20-2x. This is not right, you are. It would be 0=20+2x.

Answer 2

Start with the profit eqn
P= (25 +14x) - (5 +12x)
P=25+14x-5-12x
P=20+2x there is no breakeven point according to this model. Break even occurs when P=0 but even for x=0
P=20 This is highly unrealistic, but that the model you gave. Realistically, income doesn't have a constant term, so mayby income should just be 14x. In that case, profit would be
P=14x-5-12x
P=2x-5 breakeven when P=0
0=2x-5 add 5 to each side
2x=5 divide both sides be 2
x=2.5 to avoid loss, x>2 or x≥3 since you can't sell ½ of a shirt.

Answer 3

5+12x =25+14x
x = -10
This says your expenses were 5+12(-10) = -105
and income was 25+14(-10) = - 115

Your equations show that when x = 0 the are 5 expense units.
and when x = 0 there are 25 income units.

Therefore without making a single shirt, it has cost you 5 units and you have made 25 units for a profit of 20 units!!
The bigger x gets the more mone you make. The two lines representing Expenses never intersect at a positive value of x. Therefore there is only one possibility: Your equations are wrong. They do not reflect a real world possibility.

Answer 4

You have the set up correct. When you set this equal to zero and solve for x you do get -10. This is because your income in this case is ALWAYS GREATER than your expenses:

25 + 14x > 5 + 12x
20 + 14x > 12x
20 + 2x > 0
2x > -20
x > -10
This is always true, since x has to be positive. Even if you sell zero shirts, the income is still 25 + 14(0) = 25 while your expenses were only 5+12(0) = 5. This is a good business!