I need help in seeing if I solved these inequality problems correctly.

1. A store sells two models of computers. Because of the demand, the store stocks at least twice as many units of model A as of model B. The costs to the store for the two models are $800 and $1200, respectively. The management does not want more than $20000 in computer inventory at any one time, and it wants at least four model A and two model B computers in inventory at all times. Find and graph a system of inequalities describing all possible inventory levels.

I came up with this:

x = units of model A
y = units of model B

y <= 2x
800x + 1200y <= 20,000
x >= 4
y >= 2

2. For a concert event, there are $30 reserved seat tickets and $20 general admission tickets. There are 2000 reserved seats available, and fire regulations limit the number of paid ticket holders to 3000. the promoter must take in at $75000 in ticket sales. Find and graph a system of inequalities describing the different numbers of tickets that can be sold.

I came up with this:

x = number of reserved tickets
y = numbers of general tickets

30x <= 2000
20y <= 3000
30x + 20y <= 75000

1--2(800A + 1200B = 20,000

2--1600A + 1200B = 20,000
3--4A + 3B = 50
4--Dividing through by the smallest coefficient yields A+ A/3+B = 16+2/3
5--(A - 2)/3 must be an integer k making A = 3k + 2
6--Substituting back into (3) yields B = 14 - 4k
7--k....0....2....3....4....5
...A....2....5....8...11...14
...B...14...10....6...2....-2

Therefore, there are 4 possible inventory quantities.

For the first problem, you have the correct system of inequalities:

y ≤ 2x
800x + 1200y ≤ 20,000
x ≥ 4
y ≥ 2

To graph these inequalities, you can start by graphing the lines y = 2x, y = 2, and x = 4 on a coordinate plane.

Then, to determine the feasible region, shade the region in the coordinate plane that satisfies all the given conditions:

- The region below the line y = 2x.
- The region to the left of the line x = 4.
- The region above the line y = 2.
- The region above the line 800x + 1200y = 20,000.

For the second problem, your system of inequalities is correct:

30x ≤ 2000
20y ≤ 3000
30x + 20y ≤ 75000

To graph these inequalities, you can start by graphing the lines 30x = 2000, 20y = 3000, and 30x + 20y = 75,000 on a coordinate plane.

Then, shade the region in the coordinate plane that satisfies all the given conditions:

- The region below the line 30x = 2000.
- The region below the line 20y = 3000.
- The region below the line 30x + 20y = 75,000.

Keep in mind that in both cases, the inequalities use the symbols ≤ or ≥, so the lines representing the inequalities should be solid lines, not dashed.

For the first problem, you correctly set up the following inequalities:

1. y ≤ 2x (since the store stocks at least twice as many units of model A as model B)
2. 800x + 1200y ≤ 20,000 (to ensure the total cost of inventory doesn't exceed $20,000)
3. x ≥ 4 (to ensure there are at least four model A computers in inventory)
4. y ≥ 2 (to ensure there are at least two model B computers in inventory)

To graph these inequalities, you can plot a coordinate grid and shade the region that satisfies all the inequalities. The region that is shaded will represent the possible inventory levels.

For the second problem, you also correctly set up the following inequalities:

1. 30x ≤ 2000 (since there are only 2000 reserved seats available)
2. 20y ≤ 3000 (as fire regulations limit the number of paid ticket holders to 3000)
3. 30x + 20y ≤ 75000 (to ensure the total ticket sales reach at least $75,000)

Similarly, you can graph these inequalities by plotting a coordinate grid and shading the region that satisfies all the inequalities. The shaded region will represent the possible numbers of tickets that can be sold.

If you want to check if your solutions are correct, substitute different values for x and y that satisfy the inequalities and see if they work. For example, in the first problem, you can try substituting x = 4 and y = 2 and see if all the inequalities hold true.