Question

The area of a duck enclosure is 300 square feet, with 100 square feet occupied by a pond. Each duck in the enclosure needs more than 20 square feet of space on dry land. If x ducks can be put in the enclosure, which is the simplest inequality that represents this situation?
x < 6
x < 10
x < 15
x < 20

Answer 1

Answer: Choice B) x < 10

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Explanation:

There's 300 square feet total (land+water). Of that, 100 sq ft is taken up by the pond, so we have 300-100 = 200 sq ft of dry land. 

1 duck needs more than 20 sq ft of dry land
2 ducks need more than 40 sq ft of dry land (note how 2*20 = 40)
3 ducks need more than 60 sq ft of dry land (note how 3*20 = 60)
and so on...

In general, 
x ducks need more than 20*x square feet of dry land

The max amount of dry land we have is 200 sq ft
The total amount of space the ducks take up must be less than 200 sq ft

In terms of symbols, we have
T = total space ducks take up on dry land
T = 20*x
T < 200

So we can arrive at this inequality
20*x < 200
after replacing the "T" with "20*x"

Afterward, divide both sides by 20
20*x < 200
20*x/20 < 200/20
x < 10
So we must have less than 10 ducks

If we had 10 ducks exactly, then they would take up more than 20*10 = 200 sq ft, but we only have 200 sq ft

The largest number of ducks we can have is 9 ducks.
So that's why we have a "less than" symbol instead of a "less than or equal to" symbol.