In a manufacturing company,the cost and revenue functions of a product are given by $C(x) = 500 + \frac{5}{2}x$ and $R(x) = 3x$,where $x$ is the number of items produced and sold. How many items must be sold to realize no profit or loss? Is there any profit?

(N/A) To realize no profit or loss,the revenue must equal the cost,i.e.,$R(x) = C(x)$.
Substituting the given functions: $3x = 500 + \frac{5}{2}x$.
Subtracting $\frac{5}{2}x$ from both sides: $3x - 2.5x = 500$.
$0.5x = 500$.
$x = \frac{500}{0.5} = 1000$.
Thus,$1000$ items must be sold to realize no profit or loss.
Profit occurs when $R(x) > C(x)$,which implies $3x > 500 + 2.5x$,or $0.5x > 500$,which means $x > 1000$. Therefore,there is a profit if more than $1000$ items are sold.