To find that value we need to solve the quadratic inequality. The key word, at means the profit should be greater than or equal to 388. We set up our inequality as follows:
[tex]-0.5x^2+36x-134\ge 388[/tex]
Grouping all terms on the Right Hand Side gives
[tex]-0.5x^2+36x-522\ge 0[/tex]
We now use the quadratic formular to obtain the factors of the quadratic trinomial,
[tex](x-(6\sqrt{7} +36)(x-(36-6\sqrt{7}))\ge 0[/tex]
At this point, we use the number line to obtain the solution of this quadratic inequality. (see attachment)
By testing the 3 regions, we obtain the truth set as follows;
[tex]x\le 6\sqrt{7} +36 \: or\: x\ge 36-6\sqrt{7}[/tex]
Hence the smallest amount in dollars he should charge is,
[tex]36-6\sqrt{7}=20[/tex] to nearest dollar
