Students solve inequalities to determine how many toppings they can afford on a pizza.

Let’s solve this step by step.

We are building a pizza with toppings, and we need to write inequalities based on the rules given. The goal is to find out how many of each topping we can add without going over budget or breaking any rules.

First, let’s list all the prices:

Crust:
- Large = $15.00
- Medium = $12.50
- Small = $8.50

Toppings (per unit):
- Pepperoni = $3.50
- Pineapple = $3.75
- Bacon = $4.50
- Tomatoes = $2.25
- Peppers = $1.75
- Onions = $3.75
- Cheese = $1.50

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Now, let’s look at the rules:

> “You have $30 to spend on your pizza. You must choose one crust size. You can only pick 3 meat toppings. The pizza place has a $1 discount for every vegetable topping you add.”

Also, note:
- Meat toppings: Pepperoni, Bacon → (Pineapple? Wait — pineapple is fruit, not meat. Let’s check again.)

Actually, looking at the images:

Meat toppings (probably):
→ Pepperoni (sausage-looking), Bacon (strips)
Vegetable toppings:
→ Tomatoes, Peppers, Onions, Pineapple? (Wait — pineapple is fruit, but maybe they count it as veggie here? Or maybe not.)

But in the problem, it says:

> “You can only pick 3 meat toppings.” → So likely, meat = pepperoni, bacon. What about pineapple? Probably NOT meat. Let’s assume:

Meat toppings: Pepperoni, Bacon
Veggie toppings: Tomatoes, Peppers, Onions, Pineapple? Hmm… actually, in real life, pineapple is fruit, but sometimes pizzerias group it with veggies for pricing. But let’s read carefully.

Wait — the problem doesn’t explicitly say which are meat or veggie. But from common sense and typical pizza menus:

- Meat: Pepperoni, Bacon
- Veggie: Tomatoes, Peppers, Onions, Pineapple (sometimes considered veggie in pizza context), Cheese? No — cheese is dairy, probably not counted as veggie for discount.

BUT — the rule says: “$1 discount for every vegetable topping you add”

So we need to know which toppings qualify as “vegetable” for the discount.

Looking back at the image labels:

The toppings shown are:

Pepperoni, Pineapple, Bacon, Tomatoes, Peppers, Onions, Cheese.

In most school problems like this, they consider:

- Meat: Pepperoni, Bacon
- Veggie: Tomatoes, Peppers, Onions, Pineapple (yes, often grouped as veggie in these contexts)
- Cheese: Usually separate — not meat, not veggie? Or maybe it’s considered veggie? But let’s see the discount rule.

Actually, the problem says: “$1 discount for every vegetable topping you add”

And in the example calculation below, they mention:

> “I chose large crust ($15). I added 2 pepperonis ($7), 1 bacon ($4.50), 1 tomato ($2.25), 1 pepper ($1.75), 1 onion ($3.75). Total before discount: $15 + $7 + $4.50 + $2.25 + $1.75 + $3.75 = $34.25. Then subtract $3 discount (for 3 veggies: tomato, pepper, onion) → $31.25. Still over $30.”

Ah! So in their example, they treated tomato, pepper, onion as veggies → got $3 discount.

They did NOT include pineapple or cheese in that example.

Also, they had 2 pepperonis and 1 bacon → that’s 3 meat toppings → allowed.

So let’s define clearly:

✔ Meat toppings: Pepperoni, Bacon
✔ Veggie toppings: Tomatoes, Peppers, Onions, Pineapple? In the example, they didn’t use pineapple, but let’s assume pineapple counts as veggie too — since it’s not meat.
✘ Cheese: Probably not counted as veggie for discount — because in the example, they didn’t get discount for cheese, and cheese is listed separately.

Wait — in the example, they didn’t add cheese. Let’s check if cheese is considered veggie.

Actually, in the problem statement, it doesn’t specify, but in the sample calculation, they only discounted for tomato, pepper, onion — so likely, those three are the main veggies. But pineapple might also be included.

To avoid confusion, let’s go by what’s standard in such problems:

Typically:

- Meat: Pepperoni, Bacon
- Veggie: Tomatoes, Peppers, Onions, Pineapple
- Cheese: Not counted as veggie for discount — it’s a base ingredient or separate.

But wait — in the price list, cheese is $1.50, same as others. And in the example, they didn’t include cheese, so maybe we can choose whether to add it or not.

Actually, the problem doesn’t force us to add cheese — it’s optional.

So for our solution, we’ll assume:

Meat toppings (max 3): Pepperoni, Bacon
Veggie toppings (give $1 discount each): Tomatoes, Peppers, Onions, Pineapple
Cheese: Optional, no discount, not meat.

But let’s confirm with the example:

Example:
Crust: Large $15
Toppings: 2 pepperoni ($3.50 x 2 = $7), 1 bacon ($4.50), 1 tomato ($2.25), 1 pepper ($1.75), 1 onion ($3.75)
Total before discount: 15 + 7 + 4.50 + 2.25 + 1.75 + 3.75 = let’s calculate:

15 + 7 = 22
22 + 4.50 = 26.50
26.50 + 2.25 = 28.75
28.75 + 1.75 = 30.50
30.50 + 3.75 = 34.25 → yes.

Discount: 3 veggies (tomato, pepper, onion) → $3 off → 34.25 - 3 = 31.25 > 30 → too much.

So they want us to stay under or equal to $30 after discount.

Now, the task is to build your own pizza within $30, following the rules.

But the user didn’t ask a specific question — they just uploaded the worksheet. However, since the instruction says “solve the problem accurately”, and the worksheet has sections A, B, C, D, E — probably we need to complete them.

Looking at the bottom part:

There are boxes labeled A, B, C, D, E with questions.

Let me transcribe them:

A. Write an inequality that represents the total cost of your pizza being less than or equal to $30.

B. If you choose medium crust, how much money do you have left for toppings after buying the crust?

C. List two different combinations of toppings that would keep your total cost under $30. Show your work.

D. If you want to maximize the number of toppings, which crust should you choose? Why?

E. Write a sentence explaining how the $1 discount per vegetable topping affects your choices.

Okay, so we need to answer A through E.

Let’s start with A.

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A. Write an inequality that represents the total cost of your pizza being less than or equal to $30.

Let’s define variables.

Let:

- C = cost of crust (either 15, 12.50, or 8.50)
- P = number of pepperoni toppings
- B = number of bacon toppings
- T = number of tomato toppings
- Pe = number of pepper toppings
- O = number of onion toppings
- Pi = number of pineapple toppings
- Ch = number of cheese toppings

But we have constraints:

- Max 3 meat toppings: P + B ≤ 3
- Veggie toppings give $1 discount each: V = T + Pe + O + Pi (assuming pineapple is veggie)
- Discount = V dollars
- Total cost = C + 3.50P + 4.50B + 2.25T + 1.75Pe + 3.75O + 3.75Pi + 1.50Ch - V ≤ 30

But V = T + Pe + O + Pi, so:

Total cost = C + 3.50P + 4.50B + 2.25T + 1.75Pe + 3.75O + 3.75Pi + 1.50Ch - (T + Pe + O + Pi) ≤ 30

Simplify the veggie terms:

For tomatoes: 2.25T - T = 1.25T
For peppers: 1.75Pe - Pe = 0.75Pe
For onions: 3.75O - O = 2.75O
For pineapple: 3.75Pi - Pi = 2.75Pi

So total cost = C + 3.50P + 4.50B + 1.25T + 0.75Pe + 2.75O + 2.75Pi + 1.50Ch ≤ 30

And constraint: P + B ≤ 3

This is the inequality.

But perhaps for simplicity, since the problem might expect a general form, we can write:

Let S = sum of costs of all toppings chosen, minus $1 for each veggie topping, plus crust cost ≤ 30.

But to be precise, let's write it as:

Crust_cost + Σ(topping_costs) - (number_of_veggie_toppings) ≤ 30

With the constraint that number_of_meat_toppings ≤ 3.

Since the problem says "write an inequality", we can write:

Let M = number of meat toppings (pepperoni + bacon), V = number of veggie toppings (tomatoes, peppers, onions, pineapple), and let T_cost be the total cost of all items before discount.

Then:

T_cost - V ≤ 30

But T_cost includes crust and all toppings.

More accurately:

Let C be crust cost.

Let P = qty pepperoni, B = qty bacon, T = qty tomatoes, Pe = qty peppers, O = qty onions, Pi = qty pineapple, Ch = qty cheese.

Then:

C + 3.50P + 4.50B + 2.25T + 1.75Pe + 3.75O + 3.75Pi + 1.50Ch - (T + Pe + O + Pi) ≤ 30

And P + B ≤ 3

We can simplify as above.

But for the answer, perhaps they want a simpler version.

Notice that in the example, they calculated total before discount, then subtracted discount.

So inequality could be:

(Crust cost + sum of all topping costs) - (number of veggie toppings) ≤ 30

With the understanding that meat toppings are limited to 3.

So for part A, we can write:

Inequality:

Let \( c \) = cost of crust
Let \( m \) = total cost of meat toppings (pepperoni and bacon)
Let \( v \) = total cost of veggie toppings (tomatoes, peppers, onions, pineapple)
Let \( ch \) = cost of cheese (if added)
Let \( n_v \) = number of veggie toppings

Then:
\[ c + m + v + ch - n_v \leq 30 \]
and
\[ \text{number of meat toppings} \leq 3 \]

But to make it numerical, perhaps better to use the simplified coefficients.

From earlier simplification:

Total cost = C + 3.50P + 4.50B + 1.25T + 0.75Pe + 2.75O + 2.75Pi + 1.50Ch ≤ 30

With P + B ≤ 3

I think this is acceptable.

But let's move to B, which is easier.

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B. If you choose medium crust, how much money do you have left for toppings after buying the crust?

Medium crust = $12.50

Total budget = $30

So money left for toppings (before any discount) = 30 - 12.50 = $17.50

But wait — the discount applies when you add veggie toppings, so the actual amount you can spend on toppings before discount might be more, but the question says: "how much money do you have left for toppings after buying the crust?"

It probably means after paying for crust, how much is left from the $30 to spend on toppings, before considering discount.

Because the discount reduces the total cost, but the initial allocation is 30 - crust.

In the example, they had large crust $15, so 30-15=15 left for toppings, but they spent 19.25 on toppings before discount, then got discount.

The question is: "how much money do you have left for toppings after buying the crust?"

I think it's simply 30 - crust cost.

So for medium crust: 30 - 12.50 = 17.50

Answer: $17.50

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C. List two different combinations of toppings that would keep your total cost under $30. Show your work.

We need to create two different sets of toppings (with crust choice) such that total cost ≤ 30.

Let’s choose small crust to save money: $8.50

Then we have 30 - 8.50 = 21.50 to spend on toppings before discount, but after discount it will be less.

Actually, total cost = crust + toppings cost - discount ≤ 30

So let’s try combination 1:

Small crust: $8.50

Add 1 pepperoni ($3.50), 1 bacon ($4.50) → meat: 2, ok

Add 1 tomato ($2.25), 1 pepper ($1.75), 1 onion ($3.75) → veggies: 3, so discount $3

Add 1 cheese ($1.50)

Total before discount: 8.50 + 3.50 + 4.50 + 2.25 + 1.75 + 3.75 + 1.50

Calculate:

8.50 + 3.50 = 12.00
12.00 + 4.50 = 16.50
16.50 + 2.25 = 18.75
18.75 + 1.75 = 20.50
20.50 + 3.75 = 24.25
24.25 + 1.50 = 25.75

Discount: 3 veggies → $3 off

Total: 25.75 - 3 = 22.75 ≤ 30 → good.

Combination 1: Small crust, 1 pepperoni, 1 bacon, 1 tomato, 1 pepper, 1 onion, 1 cheese → total $22.75

Now combination 2: different.

Try medium crust: $12.50

Add 2 pepperoni ($7.00), 1 bacon ($4.50) → meat: 3, ok

Add 2 tomatoes ($4.50), 1 pepper ($1.75) → veggies: 3, discount $3

No cheese.

Total before discount: 12.50 + 7.00 + 4.50 + 4.50 + 1.75 = ?

12.50 + 7 = 19.50
19.50 + 4.50 = 24.00
24.00 + 4.50 = 28.50
28.50 + 1.75 = 30.25

Discount: 3 veggies → $3 off → 30.25 - 3 = 27.25 ≤ 30 → good.

Combination 2: Medium crust, 2 pepperoni, 1 bacon, 2 tomatoes, 1 pepper → total $27.25

We could add cheese or more, but this is fine.

So two combinations:

1. Small crust, 1 pepperoni, 1 bacon, 1 tomato, 1 pepper, 1 onion, 1 cheese → $22.75

2. Medium crust, 2 pepperoni, 1 bacon, 2 tomatoes, 1 pepper → $27.25

Both under $30.

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D. If you want to maximize the number of toppings, which crust should you choose? Why?

To maximize number of toppings, we want to spend as little as possible on crust, so we have more money for toppings.

Smallest crust is small: $8.50

Then we have 30 - 8.50 = 21.50 to spend on toppings before discount, but since veggies give discount, we can afford more.

Each veggie topping effectively costs less because of the $1 discount.

For example, a tomato costs $2.25, but with discount, net cost is $1.25.

Similarly, pepper: $1.75 - $1 = $0.75 net

Onion: $3.75 - $1 = $2.75 net

Pineapple: $3.75 - $1 = $2.75 net

Cheese: $1.50, no discount

Meat: pepperoni $3.50, bacon $4.50, no discount, and max 3.

To maximize number of toppings, we should choose cheap toppings and use the discount.

Cheapest toppings after discount:

- Pepper: $0.75 net (since $1.75 - $1 discount)

- Tomato: $1.25 net

- Cheese: $1.50 (no discount)

- Onion: $2.75 net

- Pineapple: $2.75 net

- Pepperoni: $3.50

- Bacon: $4.50

So best to add as many peppers and tomatoes as possible, since they are cheapest after discount.

Also, we can add up to 3 meat toppings, but they are expensive, so to maximize count, maybe add only 1 or 2 meat, or even none? But the rule is "you can only pick 3 meat toppings" — it doesn't say you must pick any. So we can pick 0 meat toppings.

Is that allowed? The rule says "you can only pick 3 meat toppings", which implies maximum 3, but minimum 0.

In the example, they picked 3, but we can pick fewer.

To maximize number of toppings, we should minimize cost per topping, so avoid expensive meats.

So let’s try with small crust: $8.50

Budget for toppings before discount: but total cost must be ≤30, so after discount.

Let V = number of veggie toppings, each gives $1 discount.

Let M = number of meat toppings, 0≤M≤3

Let Ch = number of cheese

Total cost = 8.50 + cost_of_meat + cost_of_veggies + cost_of_cheese - V ≤ 30

Cost_of_veggies depends on which veggies.

To minimize cost, choose cheapest veggies: peppers and tomatoes.

Pepper: $1.75 each, net $0.75 after discount

Tomato: $2.25 each, net $1.25 after discount

So pepper is cheaper.

Also, cheese: $1.50, no discount, so net $1.50

Compare to pepper net $0.75, so pepper is better.

So to maximize count, add as many peppers as possible.

But we have to pay for them before discount, but the discount reduces total.

Set up:

Let P_pep = number of peppers

Each pepper costs $1.75, but gives $1 discount, so net addition to cost is $0.75 per pepper.

Similarly, if we add a tomato, net $1.25

But since we want to maximize count, and pepper has lowest net cost, we should add only peppers and maybe cheese or tomatoes if needed.

But we can add multiple of same topping? The problem doesn't say we can't. In the example, they added 2 pepperoni, so probably we can add multiple of same topping.

So let’s assume we can add multiple units of same topping.

So for small crust: $8.50

Add M meat toppings: to minimize cost, add 0 meat.

Add V veggie toppings: all peppers, since cheapest net cost.

Each pepper: cost $1.75, discount $1, so net $0.75 per pepper.

But the discount is applied per veggie topping, so if we add V peppers, discount is V dollars.

Total cost = 8.50 + 1.75 * V - V = 8.50 + 0.75V ≤ 30

So 0.75V ≤ 21.50

V ≤ 21.50 / 0.75 = 28.666...

So V ≤ 28

Total cost = 8.50 + 0.75*28 = 8.50 + 21.00 = 29.50 ≤ 30

Number of toppings: 28 peppers

But is that realistic? Probably not, but mathematically ok.

We could add cheese: each cheese $1.50, no discount, so net $1.50, which is higher than pepper's $0.75, so better to add more peppers.

But 28 peppers might be too many, but the problem doesn't limit quantity per topping, only that meat toppings max 3.

So with small crust, 0 meat, 28 peppers: total cost 8.50 + 1.75*28 - 28 = 8.50 + 49 - 28 = 8.50 + 21 = 29.50 ≤ 30

Number of toppings: 28

If we add one cheese: cost increases by 1.50, no additional discount, so total cost 29.50 + 1.50 = 31.00 > 30, too much.

So 28 peppers is max with small crust.

But can we do better with other crusts? No, because larger crust costs more, leaving less for toppings.

For example, medium crust $12.50, then 0.75V ≤ 30 - 12.50 = 17.50, V ≤ 17.50 / 0.75 ≈ 23.33, so V=23, cost=12.50 + 0.75*23=12.50+17.25=29.75, number of toppings=23 < 28.

Similarly, large crust worse.

So small crust allows most toppings.

But is there a way to get more than 28?

What if we add some tomatoes? Tomato net cost $1.25, which is higher than pepper's $0.75, so worse for maximizing count.

Or add meat? Meat has no discount, pepperoni $3.50, which is high, so adding meat would reduce the number of toppings we can add.

For example, if we add one pepperoni ($3.50), then for small crust, total cost = 8.50 + 3.50 + 0.75V ≤ 30 → 12.00 + 0.75V ≤ 30 → 0.75V ≤ 18 → V≤24, total toppings=1+24=25<28.

Worse.

So best is small crust, 0 meat, 28 peppers, total cost $29.50, 28 toppings.

But perhaps the problem expects reasonable numbers, but mathematically, this is correct.

We could add cheese instead of some peppers, but cheese net $1.50 > $0.75, so worse.

Another idea: add pineapple? Net cost $2.75, worse.

Or onions $2.75, worse.

So peppers are best.

But in reality, you might not want 28 peppers, but for the problem, it's fine.

Perhaps the discount is only for distinct types? But the problem doesn't say that. In the example, they added one of each veggie, but didn't specify if multiple of same type count.

The rule says: "$1 discount for every vegetable topping you add" — "every" suggests per item, not per type.

In the example, they added one tomato, one pepper, one onion — three items, got $3 discount.

If they added two tomatoes, probably get $2 discount for tomatoes.

So yes, per item.

So 28 peppers is valid.

But to be practical, perhaps list a reasonable combination.

The question is to maximize number, so 28 is the answer.

But let's confirm calculation:

Small crust: $8.50

28 peppers: 28 * $1.75 = $49.00

Discount: 28 * $1 = $28.00

Total: 8.50 + 49.00 - 28.00 = 8.50 + 21.00 = 29.50 ≤ 30

Yes.

Number of toppings: 28

If we try 29 peppers: 29*1.75=50.75, discount 29, total=8.50+50.75-29=8.50+21.75=30.25>30, too much.

So max 28.

Thus, choose small crust to maximize number of toppings.

Answer: Small crust, because it leaves the most money for toppings, and with the discount on veggie toppings, you can afford many low-cost toppings like peppers.

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E. Write a sentence explaining how the $1 discount per vegetable topping affects your choices.

The discount makes vegetable toppings cheaper, so I am more likely to choose them over meat or cheese to save money and stay within budget.

Or: It encourages me to add more vegetable toppings since each one reduces the total cost by $1, allowing me to add more toppings or spend less.

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Now, to summarize answers:

A. Inequality: Let C be crust cost, P number of pepperoni, B number of bacon, T number of tomatoes, Pe number of peppers, O number of onions, Pi number of pineapple, Ch number of cheese. Then:

C + 3.50P + 4.50B + 2.25T + 1.75Pe + 3.75O + 3.75Pi + 1.50Ch - (T + Pe + O + Pi) ≤ 30

With P + B ≤ 3

Simplified: C + 3.50P + 4.50B + 1.25T + 0.75Pe + 2.75O + 2.75Pi + 1.50Ch ≤ 30

B. $17.50

C. Two combinations:

1. Small crust ($8.50), 1 pepperoni ($3.50), 1 bacon ($4.50), 1 tomato ($2.25), 1 pepper ($1.75), 1 onion ($3.75), 1 cheese ($1.50). Total before discount: $25.75, discount $3 (for 3 veggies), total $22.75 ≤ 30.

2. Medium crust ($12.50), 2 pepperoni ($7.00), 1 bacon ($4.50), 2 tomatoes ($4.50), 1 pepper ($1.75). Total before discount: $30.25, discount $3 (for 3 veggies), total $27.25 ≤ 30.

D. Small crust, because it costs the least, leaving more money for toppings. With the discount on vegetable toppings, you can add many low-cost toppings like peppers, maximizing the total number.

E. The $1 discount per vegetable topping makes them more affordable, so I prefer to add vegetables to reduce the total cost and fit more toppings within the $30 budget.

Now, for the final answer, since the user didn't specify which part, but the worksheet has A-E, probably we need to provide all.

But the instruction says "provide only the final correct answer" in Final Answer section, but that might be for a single answer. However, since there are multiple parts, perhaps list them.

Looking back at the user's message: "Solve the problem accurately." and the image has the whole worksheet.

In the response format, it says "Final Answer: [Provide only the final correct answer]", but for multi-part, perhaps we need to box all.

To be safe, I'll provide the answers for A-E in the Final Answer section.

But let's see the options: ExplainAnswer is false, etc., so no extra sections.

So in Final Answer, I'll write the answers concisely.

Perhaps:

Final Answer:
A. \( C + 3.50P + 4.50B + 1.25T + 0.75Pe + 2.75O + 2.75Pi + 1.50Ch \leq 30 \) with \( P + B \leq 3 \)
B. $17.50
C. Combination 1: Small crust, 1 pepperoni, 1 bacon, 1 tomato, 1 pepper, 1 onion, 1 cheese ($22.75); Combination 2: Medium crust, 2 pepperoni, 1 bacon, 2 tomatoes, 1 pepper ($27.25)
D. Small crust, because it minimizes crust cost, allowing more money for toppings, and with vegetable discounts, you can add many inexpensive toppings like peppers.
E. The discount makes vegetable toppings cheaper, encouraging their selection to stay within budget and add more toppings.

But for C, I used specific numbers, but in combination 2, I have 2 tomatoes and 1 pepper, which is 3 veggies, discount $3.

In combination 1, 3 veggies.

Both valid.

For D, I said 28 toppings, but perhaps they expect the crust choice, not the number.

The question is "which crust should you choose? Why?" so answer is small crust, with reason.

Similarly for others.

I think this is fine.

So let's write the Final Answer.