Michelle has $9 and wants to buy a combination of dog food to feed at least two dogs at the animal shelter. A serving of dry food costs $1, and a serving of wet food costs $3.

Question

Describe the graph of the system of inequalities, including shading and the types of lines graphed. Provide a description of the solution set

Answer 1

The system of inequalities can be written as:

x + y ≥ 2

where x represents the number of servings of dry food and y represents the number of servings of wet food.

The total cost of the dog food can also be expressed as:

C = x + 3y

To graph the system of inequalities, we can plot the line x + y = 2 with a solid line since it includes the points where there are exactly 2 servings of dog food. We will shade the region above and to the right of the line, since any point in that region corresponds to a combination of dry and wet food servings that add up to 2 or more.

Next, we will plot the line C = 9 (representing the total cost of the dog food) by rearranging the equation to y = (9-x)/3, which has a slope of -1/3 and a y-intercept of (0,3). We will draw a dashed line since it does not include any valid solutions, which must be in the region defined by the first equation. We will shade the area below the line since any point below the line represents a combination of x and y servings whose total cost is less than or equal to $9.

The solution set consists of all the points that satisfy both inequalities, which is the shaded region in the upper right quadrant bounded by the solid line x + y = 2 and the dashed line C = 9. Any combination of dry and wet food servings within that shaded region will provide enough food for at least two dogs, and will cost $9 or less.