Answer
$400\leq x \leq 4400$
Work Step by Step
We have to find interval of the following inequality: $Profit \geq 2400$
$Profit = revenue - cost$ $=$ $20x-(2000+8x+0.0025x^2)$
Let the $Profit$ be $P$.
$P=20x-2000-8x-0.0025x^2=-0.0025x^2+12x-2000$
So, we have to find interval of the inequality:
$-0.0025x^2+12x-2000\geq 2400$
$-0.0025x^2+12x-4400\geq 0$ (Multiply by -400)
$x^2-4800x+1760000\leq0$
Now, we can write it as a quadratic equation and solve it using interval notation
$x^2-4800x+1760000=0$
$D=b^2-4ac=16,000,000$
$x=\frac{-b±\sqrt{D}}{2a}=\frac{4800±4000}{2}$
$x_1=400$ ; $x_2=4400$
We have $3$ possible intervals:
$(-\infty, 400]$ - Positive
$[400, 4400]$ - Negative
$[4400, +\infty)$ - Positive
Now we get back to the last inequality. We needed interval less than or equal to $0$, so the second interval is appropriate: $[400, 4400]$
$400\leq x \leq 4400$
