Michael has $8 and wants to buy a combination of cupcakes and fudge to feed at least four siblings. Each cupcake costs $2, and each piece of fudge costs $1.
This system of inequalities models the scenario:
2x + y ≤ 8
x + y ≥ 4
Describe the graph of the system of inequalities, including shading and the types of lines graphed. Provide a description of the solution set
The first inequality, 2x + y ≤ 8, can be rewritten in slope-intercept form as y ≤ -2x + 8. This inequality is a linear equation with a slope of -2 and a y-intercept of 8. The line is solid (not dashed) because the inequality includes the equal sign. Shade the area below the line to represent all the possible combinations of cupcakes and fudge that Michael can afford with $8.
The second inequality, x + y ≥ 4, can be rewritten in slope-intercept form as y ≥ -x + 4. This inequality is a linear equation with a slope of -1 and a y-intercept of 4. The line is solid (not dashed) because the inequality includes the equal sign. Shade the area above the line to represent all the possible combinations of cupcakes and fudge that feed at least four siblings.
The intersection of the two shaded regions is the solution set of the system of inequalities. This represents all the possible combinations of cupcakes and fudge that Michael can buy to feed at least four siblings, without spending more than $8.
To describe the graph of the system of inequalities, we first look at each inequality separately.
The first inequality, 2x + y ≤ 8, can be rewritten as y ≤ -2x + 8. This inequality represents a line with a slope of -2 and a y-intercept of 8. The line will be solid because y is less than or equal to the expression -2x + 8.
The second inequality, x + y ≥ 4, can be rewritten as y ≥ -x + 4. This inequality represents a line with a slope of -1 and a y-intercept of 4. The line will also be solid because y is greater than or equal to the expression -x + 4.
Now, let's graph these two lines on a coordinate plane.
First, let's graph the line y = -2x + 8. We can plot a few points to make the process easier.
When x = 0, y = 8.
When x = 4, y = 0.
Plot these points and draw a straight line through them. Label this line as line A.
Next, let's graph the line y = -x + 4. Again, plot a few points.
When x = 0, y = 4.
When x = 4, y = 0.
Plot these points and draw a straight line through them. Label this line as line B.
Now, for the shading and description of the solution set:
Since we have y ≤ -2x + 8 and y ≥ -x + 4, we can shade the region below line A (y = -2x + 8) and above line B (y = -x + 4). The shaded region represents all the possible combinations of cupcakes and fudge that Michael can buy within his budget of $8.
To describe the solution set, we can say that it consists of the points below line A and above line B, including any points on the lines themselves.
Overall, the solution set represents the combinations of cupcakes and fudge that Michael can purchase to feed at least four siblings while staying within his budget.