14. A café sells peach smoothies and berry smoothies. The café makes a profit of \( \$ 2.25 \) for each peach smoothie that is sold and a profit of \( \$ 2 \) for each berry smoothie that is sold. The owner of the café wants to make a total profit of more than \( \$ 90 \) per day from the sales of smoothies. a. Write an inequality that represents the number of peach smoothies \( p \) and berry smoothies \( b \) that the café needs to sell. Describe the constraints on the variables. b. Graph the solution of the inequality on a coordinate plane. c. On Monday, the café sold 20 peach smoothies and made the daily profit goal. What can you say about the number of berry smoothies that were sold on Monday?

**a. Setting Up the Inequality** The profit from selling \( p \) peach smoothies is \( \$2.25 \) each, contributing a total of \( 2.25p \) dollars to the profit. The profit from selling \( b \) berry smoothies is \( \$2 \) each, contributing \( 2b \) dollars. The café’s daily profit must be more than \( \$90 \), so the inequality is \[ 2.25p + 2b > 90. \] Since it is not possible to sell a negative number of smoothies, we have the constraints \[ p \ge 0 \quad \text{and} \quad b \ge 0. \] --- **b. Graphing the Inequality** 1. **Identify the Boundary Line:** The boundary line of the inequality is found by replacing the \( > \) sign with \( = \): \[ 2.25p + 2b = 90. \] 2. **Find the Intercepts of the Boundary Line:** - When \( p = 0 \): \[ 2b = 90 \quad \Longrightarrow \quad b = 45. \] This gives the point \((0, 45)\). - When \( b = 0 \): \[ 2.25p = 90 \quad \Longrightarrow \quad p = \frac{90}{2.25} = 40. \] This gives the point \((40, 0)\). 3. **Sketch the Graph:** - Plot the points \((0, 45)\) and \((40, 0)\) on the coordinate plane. - Draw the dashed line connecting these two points, indicating that points on the line are not included in the solution (because the inequality is strict, \(>\)). - Shade the region above the line, since testing a point (for instance, \((20, 23)\)) shows: \[ 2.25(20) + 2(23) = 45 + 46 = 91 > 90. \] Thus, the shaded region is the set of all points \((p, b)\) where the inequality holds. Remember that \( p \) and \( b \) are only meaningful for non-negative values. --- **c. Determining the Number of Berry Smoothies Sold on Monday** On Monday, the café sold \( 20 \) peach smoothies. Substitute \( p = 20 \) into the inequality: \[ 2.25(20) + 2b > 90. \] Calculate the profit from peach smoothies: \[ 2.25(20) = 45. \] Thus, the inequality becomes: \[ 45 + 2b > 90. \] Subtract \( 45 \) from both sides: \[ 2b > 45. \] Divide both sides by \( 2 \): \[ b > 22.5. \] Since the number of smoothies must be a whole number, the café must have sold at least \( 23 \) berry smoothies. --- \[ \textbf{Summary of Answers:} \] - **(a)** The inequality is \[ 2.25p + 2b > 90 \quad \text{with} \quad p \ge 0 \quad \text{and} \quad b \ge 0. \] - **(b)** The boundary line \(2.25p + 2b = 90\) has intercepts \((0, 45)\) and \((40, 0)\); the solution is the region above this dashed line. - **(c)** The café must have sold at least \( 23 \) berry smoothies on Monday.