System of Inequalities

How do you solve: Ho sells ice cream cones at the county fair. She has to rent the equipment for $48 and spend $0.53 on ingredients for each cone. Write an inequality to represent the possible numbers of ice cream cones that she must sell at $1.40 each in order to make a profit.

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Jan 16, 2011
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by: Karin

For this problem, you will want to think about writing two different equations.

The first equation is the cost to make he ice cream.

You know that it costs $48 for the equipment. This is a flat fee (therefore, it is the constant or y-intercept). You also know that it costs $0.53 for each cone.

Let y = total
Let x = number of ice cream cones

y = .53x + 48 This is the amount of money it would cost to make the ice cream cones.

Now let's think about how much we would make if we sold the ice cream cones.

Each ice cream cone is sold for $1.40. So...
y = 1.40x (This equation represents the sales proceeds.)

Now we want to write an inequality to show when Ho would make a profit.

In order to make a profit, you want your sales proceeds to be greater than your cost.

Therefore;
1.40x > .53x + 48

We've taken our two equations (cost equation and sales equation and used them to write the inequality.)

Your problem only asks for you to write the inequality, but if you need to solve it, you would need to first get your variable terms on the same side of the inequality.

1.40x - .53x > .53x - .53x + 48

.87x > 48

Now divide by .87 in order to isolate the variable.

.87x/.87 > 48/.87

x > 55.17

Since you can't sell part of an ice cream cone, we would round up.

x > 56

Let's check to see if selling 56 cones will yield a profit.

.53(56)+48 = 77.68 The cost is $77.68

Let's check the sales proceeds:
1.40(56) = 78.40 Proceeds from sales

Since the sales proceeds ($78.40) is greater than the cost ($77.68), we have correctly written the inequality that represents the profit.

I hope this helps,
Karin

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