Let's start by defining our variables and formulating the problem to identify the appropriate inequality.
1. Initial price and sales:
- Initial price of each package, [tex]\( P_{\text{initial}} = \$2.29 \)[/tex]
- Initial sales, [tex]\( S_{\text{initial}} = 95 \)[/tex] packages
2. Price and sales change with [tex]\( x \)[/tex]:
- For each 20-cent increase in price, defined by [tex]\( x \)[/tex]:
- New price per package, [tex]\( P_{\text{new}} = 2.29 + 0.20x \)[/tex]
- New sales per day, [tex]\( S_{\text{new}} = 95 - 9x \)[/tex]
3. Revenue equation:
- Revenue [tex]\( R \)[/tex] is given by the product of the new price and the new sales:
[tex]\[
R = P_{\text{new}} \times S_{\text{new}}
\][/tex]
Substituting the expressions for [tex]\( P_{\text{new}} \)[/tex] and [tex]\( S_{\text{new}} \)[/tex]:
[tex]\[
R = (2.29 + 0.20x)(95 - 9x)
\][/tex]
4. Expanding the revenue equation:
- Multiply the terms:
[tex]\[
R = 2.29 \cdot 95 + 2.29 \cdot (-9x) + 0.20x \cdot 95 + 0.20x \cdot (-9x)
\][/tex]
Simplify the expression:
[tex]\[
R = 217.55 - 20.61x + 19x - 1.8x^2
\][/tex]
Combine like terms:
[tex]\[
R = -1.8x^2 - 1.61x + 217.55
\][/tex]
5. Setting up the inequality for revenue:
- We require the revenue to be at least [tex]\( \$255 \)[/tex]:
[tex]\[
R \geq 255
\][/tex]
Substitute the revenue expression:
[tex]\[
-1.8x^2 - 1.61x + 217.55 \geq 255
\][/tex]
Given this derived inequality, the correct option among the choices is:
[tex]\[ \boxed{D. \, -1.8x^2 - 1.61x + 217.55 \geq 255} \][/tex]
