We are given:
P(x) = -5(x - 4)^2 + 45
where x = number of pairs of shoes in thousands, and P(x) = profit in thousands of dollars.
We want to find values of x for which the company earns a profit:
P(x) > 0
Set inequality
-5(x - 4)^2 + 45 > 0
Move terms
-5(x - 4)^2 > -45
Divide by -5 (reverse inequality)
(x - 4)^2 < 9
Take square root
-3 < x - 4 < 3
Solve for x
1 < x < 7
Answer:
The company will make a profit if it sells between 1,000 and 7,000 pairs of shoes.