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.
Describe the graph of the system of inequalities, including shading and the types of lines graphed. Provide a description of the solution set
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.
To solve this problem graphically, let's define some variables:
Let x be the number of servings of dry food.
Let y be the number of servings of wet food.
The cost of x servings of dry food is $1x = $x.
The cost of y servings of wet food is $3y.
Now let's form the system of inequalities based on the given conditions:
1. Michelle has $9, so the total cost should be less than or equal to $9:
x + 3y ≤ 9
2. Michelle wants to feed at least two dogs, so the total number of servings must be greater than or equal to 2:
x + y ≥ 2
To graph this system of inequalities, we will draw the lines for each equation and shade the region that satisfies both inequalities.
First, let's graph the equality line x + 3y = 9:
- Convert it into slope-intercept form: y = (-1/3)x + 3
- Plot two points on the line: (0,3) and (9,0)
- Draw a dashed line connecting the two points, representing x + 3y = 9
Next, let's graph the inequality line x + y ≥ 2:
- Convert it into slope-intercept form: y ≥ -x + 2
- Plot the boundary line y = -x + 2 (with a solid line, since it includes the inequality)
- Shade the region above the line to represent y ≥ -x + 2
The final graph would consist of a dashed line x + 3y = 9 and a solid line y = -x + 2, forming a shaded region where both inequalities are satisfied.
The solution set, represented by the shaded region, shows the values of x and y that satisfy both inequalities simultaneously. These values represent the number of servings of dry food (x) and the number of servings of wet food (y) that Michelle can buy to feed at least two dogs, given her budget of $9.