A rectangular garden measuring 13 meters by
15 meters is to have a gravel pathway of constant width built all
around it. There is enough gravel to cover 80 square meters or
less.
Enter an inequality that represents all possible widths (w), in
meters, of the pathway.

9514 1404 393

  0 ≤ w ≤ -7 +√69

Step-by-step explanation:

The dimensions in meters of the garden and its pathway are ...

  (13 +2w) by (15 +2w)

The area of the pathway is the difference between this and the area of the garden.

  path area = (13+2w)(15+2w) -13(15) = 4w² +56w

This is required to be at most 80 square meters, so we have the system of inequalities ...

  4w² +56w ≤ 80

  0 ≤ w

The solution to the quadratic can be found by completing the square.

  w² +14w ≤ 20 . . . . . divide by 4

  w² +14w +49 ≤ 69 . . . . add 7² to complete the square

  (w +7)² ≤ 69

  w +7 ≤ √69 . . . . . the negative square root is of no interest

  0 ≤ w ≤ -7 +√69 . . . . . all possible widths of the pathway

Additional comment

The solution of the quadratic, by itself, gives a range of widths from about -15.3 to 1.3 meters. We know the width cannot be negative, so we need to include the restriction w ≥ 0. This is best done at the end, after we have the solution to the quadratic. Hence our "all possible widths" expression is not just the quadratic inequality, but is the solution inequality.

correct answers below: