Marissa is a photographer. She sells framed photographs for $100 each and greeting cards for $5 each. The materials for each framed photograph cost $30, and the materials for each greeting card cost $2. Marissa can sell up to 8 framed photographs and 40 greeting cards each week, but this week, she only has $200 to spend on materials. Marissa hopes to earn a profit of at lest $400 this week after paying for her materials.

Let x = the number of framed photographs.
Let y = the number of greeting cards.
Two of the inequalities that model this situation are x < =8 and y < = 40

Explain your inequalities and explain why x<=8 and y<=40 are also inequalities for this system

I got Material cost equation : 30x + 2y = 200
Profit equation : 100x + 5y = 400

Aren't they supposed to be in-equalities?

profit = revenue - expenses
= 100x + 5y - 30x - 2y
= 70x + 3y

but
100x + 5y - 30x - 2y ≥ 400
70x + 3y ≥ 400

also 30x + 2y ≤ 200
15x + y ≤ 100

" Marissa can sell up to 8 framed photographs" --- > x ≤ 8
" and 40 greeting cards each week" ---> y ≤ 40

I would also include x≥ 0, y ≥ 0

solving the 2 linear relations as equations:
70x + 3(100-15x) = 400
70x + 300 - 45x = 400
25x = 100
x = 4
y = 40

She should make and sell 4 photographs, and 40 cards

profit+ 70(4) + 3(40) = 400

suppose she goes for 6 photographs and 40 cards ...
x=6 , y = 40
profit > 400
both x and y agree with x≤8 and y≤ 40
BUT
30(6) + 2(40) = $260
and she only wanted to spend a maximum of $200 on materials

Im confused

Marissa plans to make and sell 5 framed photographs and 25 greeting cards. Is that a solution to the system of inequalities? If so, is it the solution that will produce the most profit?

The inequality x <= 8 represents the maximum number of framed photographs Marissa can sell each week. Since she can only sell up to 8 framed photographs, the number of framed photographs cannot exceed 8. This constraint ensures that Marissa does not produce more framed photographs than she is capable of selling.

Similarly, the inequality y <= 40 represents the maximum number of greeting cards Marissa can sell each week. Since she can only sell up to 40 greeting cards, the number of greeting cards cannot exceed 40. This constraint ensures that Marissa does not produce more greeting cards than she is capable of selling.

The material cost equation, 30x + 2y = 200, represents the constraint that Marissa has a maximum budget of $200 to spend on materials. This equation ensures that the total cost of the materials used for framed photographs (30x) and greeting cards (2y) does not exceed $200.

The profit equation, 100x + 5y = 400, represents Marissa's goal of earning a profit of at least $400 after paying for her materials. This equation ensures that the total revenue earned from selling framed photographs (100x) and greeting cards (5y) is at least $400.