This math example shows that when costs (like minimum wage) are low as in the first business model, then between 45 and 68 sandwiches are being sold per day to have a daily profit of between $60 and $130 at $5.50 per sandwich.

A way you can check this solution is by entering the profit function "3x-75" in a graphing calculator in the Graph mode, choose graph-window settings that fit the x-domain and y-range of the question, then select the trace button to trace along the line of the graph until you reach the x-value 45 (representing a number of sandwiches sold), where you will see a y-value of 60 (representing lowest daily profits in dollars) and for x-value 68 (representing a number of sandwiches sold) you will see a y-value of 130 (representing the highest daily profit in dollars).

A deli based on this profit model would break even after selling 25 sandwiches, which is 8 less than the 33 sandwiches required to break even under the 2nd business model in which employees were paid $11.50 per hour.

Step 0: Replace the x in the inequality representing the lowest sales (60) and highest sales (130) with P(x) which represents the profit function equation "3x-75," solved for in Table 0.5.

60

<

P(x)

<

130

Step 1: Insert the daily Profit function for the first model in the space inbetween the inequality representing the lowest daily sales and highest daily sales.

60

<

3x - 75

<

130

Step 2: Add 75 to both sides of the inequality and the middle term.

60 +75

<

3x +75

<

130 + 75

Step 3: Combine like terms with addition.

135

<

3x

<

205

Step 4: To get "x" by itself in the middle divide both sides and the middle term by 3.