The variables are:
x: number of portraits
y: number of landscapes
She wants to sell at least 90 drawings, that is,
x + y ≥ 90
She sells the portraits for $14 each and the landscapes for $12. She needs to earn at least $504, that is,
14x + 12y ≥ 504
To graph the inequalities, first, we need to draw the lines. We can do that by finding two points on the line.
Line: x + y = 90
x = 0
0 + y = 90
y = 90
y = 0
x + 0 = 90
x = 90
The points are (0, 90) and (90, 0)
To know what area we have to shade, we need to check if a chosen point satisfies the inequality. For example, for the point (0, 0)
0 + 0 ≥ 90
0 ≥ 90
which is false, then (0, 0) is not a solution, and we have to shade above the solid line.
Line: 14x + 12y = 504
x = 0
14(0) + 12y = 504
y = 504/12
y = 42
y = 0
14x + 12(0) = 504
x = 504/14
x = 36
The points are (0, 42) and (36, 0)
Substituting with (0, 0) into the inequality,
14(0) + 12(0) ≥ 504
0 ≥ 504
which is false, then (0, 0) is not a solution, and we have to shade above the solid line.
The solution to the system of inequalities is the region above the upper line.
If Caitlyn sells 70 portraits and 40 landscapes she will meet the minimum sales and earn a profit (the point is on the solution)
If Caitlyn sells 60 portraits and 85 landscapes she will meet the minimum sales and earn a profit (the point is on the solution)
