Question

Help with #24!!! I don't get it. Please provide a clear and brief explanation! Thank you <3

Answer 1

So hmmm the band is going to charge her 750, for the whole gig, so that's a fixed amount, regardless of how many folks show up, 2 or 2,000

the caterer, charges 2.25 per head, because, the more heads, the more plates and glasses he has to make, more work for him

now, she wants to keep the charges between 2.75 and 3.25

now.. let's see her costs

750 from  the band and 2.25 per head from the caterer, let's say "p" folks show up, so the caterer will then be charging 2.25*p or 2.25p

so her cost say C(p) = 750 + 2.25p

now, we dunno what C(p) is going to end up being, that depends on how many folks show up

now, we know "p" folks are going to show up, so, if "p" show up, the average cost will then be price * p

if the price, Natasha has in mind is 2.75, then C(p) is 2.75p
if the price, Natasha has in mind is 3.25, then C(p) is 3.25p

but whatever 750 + 2.25p is, is ok if it's greater than 2.75p
         2.75p < 750 + 2.25p

and whatever 750 + 2.25p is, is not ok if it goes over 3.25p, it has to be less

        3.23p > 750 + 2.25p

$\bf 2.75p\ \textless \ 750+2.25p\implies 2.75p-2.25p\ \textless \ 750\implies 0.5p\ \textless \ 750 \\\\\\ \cfrac{1}{2}p\ \textless \ 750\implies p\ \textless \ 2\cdot 750\implies \boxed{p\ \textless \ 1500}\\\\ -------------------------------\\\\ 3.25p\ \textgreater \ 750+2.25p\implies 3.25p-2.25p\ \textgreater \ 750\implies \boxed{p\ \textgreater \ 750}\\\\ -------------------------------\\\\ \begin{cases} p\ \textless \ 1500\\ p\ \textgreater \ 750 \end{cases}\iff 750\ \textless \ p\ \textless \ 1500\impliedby \textit{using the combined form}$