Explanation:
To graph the boundary lines of the linear inequalities, we need to first set up the inequalities based on the given information. Let x be the number of popcorn bags and y be the number of cotton candy bags. Then we have:
Chris needs at least 7 bags of popcorn, so the inequality for popcorn is x ≥ 7.
Chris has $56 to spend, so the total cost of popcorn and cotton candy cannot exceed $56, which gives the inequality 3x + 7y ≤ 56.
To graph the boundary lines, we need to first graph the lines corresponding to these two inequalities:
The line x = 7 is a vertical line passing through the point (7,0), because the only restriction on the popcorn is that Chris needs at least 7 bags.
The line 3x + 7y = 56 is a diagonal line with x-intercept 56/3 and y-intercept 8, because these are the points where the line intersects the x and y axes, respectively.
We can now plot these lines on a coordinate plane:
|
8 | x = 7
| o
7 | |
| | 3x + 7y = 56
6 | |
| o
5 |
| o
4 |
|
3 |
|
2 | o
|
1 | o
|____________________________
1 2 3 4 5 6 7 8 9 10
The shaded region below and to the left of the diagonal line represents the solution set to the inequalities, because it includes all the points where Chris can buy at least 7 bags of popcorn and stay within his budget.
Finally, we can plot the given points and identify which ones are part of the solution set:
(3, 7) is not part of the solution set because it is above the diagonal line.
(2, 2) is part of the solution set because it is below and to the left of the diagonal line.
(8, 1) is not part of the solution set because it is above the diagonal line.
(10, 2) is not part of the solution set because it is above the diagonal line.
(5, 5) is part of the solution set because it is below and to the left of the diagonal line.
Therefore, the solution set consists of the points (2, 2) and (5, 5).