nathan wants to go to a hockey game with friends. Tickets are $18, but students can get discounted tickets for $12. They want to spend no more than $80, including $8 for parking. The vehicle they will take can only hold 7 people. Write and graph a system of four inequalities modeling their possible ticket purchases. Let x = number of student tickets and y = number of regular tickets.

12x+18y+8 <= 80

x+y <= 7
x > 0
y > 0

To write a system of inequalities to model Nathan and his friends' possible ticket purchases, we need to consider the constraints given in the question:

1. The number of student tickets (x) and regular tickets (y) should be non-negative, as there cannot be a negative number of tickets.

2. The total cost of tickets and parking must be less than or equal to $80.

3. The number of people attending the game (x + y) should be less than or equal to 7, as that is the maximum capacity of their vehicle.

Now, let's break down these constraints into inequalities:

1. Non-negativity constraint:
x ≥ 0
y ≥ 0

2. Cost constraint:
12x + 18y + 8 ≤ 80

Explanation: The cost of x student tickets at $12 each, y regular tickets at $18 each, plus $8 for parking, should be less than or equal to $80.

3. Maximum capacity constraint:
x + y ≤ 7

Explanation: The total number of people attending the game, x students and y regular ticket holders, should be less than or equal to 7.

Now, let's graph these inequalities on a coordinate plane:

(Note: For simplicity, the graph will be a rough representation rather than to scale)

First, graph the non-negativity constraint:

x ≥ 0
y ≥ 0

This inequality represents the portion of the coordinate plane above or on the x-axis and to the right or on the y-axis, including the origin.

Next, graph the cost constraint:

12x + 18y + 8 ≤ 80

To graph this inequality, we need to rearrange it in slope-intercept form:
12x + 18y ≤ 72
18y ≤ -12x + 64
y ≤ (-2/3)x + (64/18)
y ≤ (-2/3)x + (32/9)

Plot this as a dashed line with a slope of -2/3 and a y-intercept of 32/9. Shade the region below this line to satisfy the inequality.

Finally, graph the maximum capacity constraint:

x + y ≤ 7

Plot this as a dashed line with a slope of -1 and a y-intercept of 7. Shade the region below this line to satisfy the inequality.

The shaded region of the graph where all three inequalities overlap represents the feasible solutions for Nathan and his friends' ticket purchases.