Question

A rectangular pen for a pet is 3ft longer that it is wide, give possible values for the width W of the pen if its area must be between 180 and 460 sqaure feet, inclusivley.

Answer 1

If area is 180
W * ( W + 3) = 180
w^2 + 3w = 180
w = 12
if area is 460,
W^2 + 3w = 460,
w = 20
12<W<20

Answer 2

so that would mean 12ft and 20 ft

Answer 3

No, the width can lie ANYWHERE between 12ft and 20ft. He tested the extreme cases here, and reasoned that a value of W somewhere between 12ft and 20ft would result in an area greater than 180 but less than 460.

If you'd like to confirm this fact, try graphing\[Area=w*(w+3)=w^2+3w\]
Look at what values of w output values that are between 180 and 460 and you should see that 12ft and 20ft indeed bound these values!

I hope that helps solidify your understanding!