Let h be the number of hot dogs sold per week and let s be the number of sodas sold per week. The team would like to sell at least 25 hot dogs and 75 sodas per week. Therefore, h must be greater than or equal to 25, and p must be greater than or equal to 75. h≥25 and s≥75 The price of one hot dog is $4. Therefore, the expression 4h represents the amount earned by selling hot dogs. Similarly, since the price of one soda is $2, the expression 2s represents the amount earned by selling sodas. We are told the goal is earning at least $300. 4h+2s≤300 We can combine the three inequalities we have written to form a system of inequalities. ⎩⎪⎪⎨⎪⎪⎧h≥25s≥754h+2s≤300
Let the horizontal axis be the h-axis and the vertical axis be the s-axis. Let's graph the inequalities one at a time.
To obtain the boundary line we replace the inequality sign with an equals sign. Inequalityh≥25Boundary Lineh=25 The line h=25 is a vertical line which passes through (25,0). Since the inequality is not strict, the line will be solid. The inequality states that h is greater than or equal to 25. Therefore, we will shade the half-plane to the right of the line.
The boundary line related to the second inequality is s=75. This is a horizontal line whose y-intercept is 75. Since s is greater than or equal to 75, we will shade the region above the line. It will be solid because the inequality is not strict.
The solution to this system of inequalities is where the three shadings overlap.
To name one possible solution, we need any point from the overlapping area. We will plot the point in the graph and name it.
As we see above, one possible solution is (50,100). In the context of the problem, it means that 50 hot dogs and 100 sodas were sold.