Inequalities Challenge: Writing & Graphing Solutions

1. A farmer has a budget of $200 to spend on seeds and fertilizer. If each bag of seeds costs $20 and each bag of fertilizer costs $30, write an inequality to represent the situation. What is the feasible region for the number of bags of seeds (s) and fertilizer (f) he can buy?

• 20s + 30f ≤ 200, s ≥ 0, f ≥ 0

• 25s + 20f ≤ 200, s ≥ 0, f ≥ 0

• 10s + 40f ≤ 200, s ≥ 0, f ≥ 0

• 15s + 25f ≤ 200, s ≥ 0, f ≥ 0

2. Sarah is planning a party and wants to spend no more than $150 on decorations and food. If decorations cost $25 each and food costs $15 per person, write an inequality for the total cost. Identify the feasible region for the number of decorations (d) and people (p) she can invite.

• 30d + 20p ≤ 150; d ≤ 5, p ≥ 0.

• 25d + 20p ≤ 150; d ≥ 0, p ≤ 10

• 25d + 15p ≤ 150; feasible region: d ≥ 0, p ≥ 0.

• 20d + 10p ≤ 150; d ≥ 0, p ≥ 0.

3. A gym charges a monthly fee of $40 plus $5 for each class attended. If a member wants to spend no more than $100 in a month, write an inequality for the number of classes (c) they can attend. What does the graph of this inequality look like?

• c ≤ 15

• c ≤ 10

• c ≤ 12

• c ≤ 8

4. A school is organizing a field trip and has a budget of $500. The bus rental costs $200 and each student ticket costs $10. Write an inequality to represent the number of students (s) that can go on the trip. What is the feasible region?

• s ≤ 30

• s ≤ 40

• s ≤ 20

• s ≤ 25

5. A store sells notebooks for $3 each and pens for $2 each. If a student has $30 to spend, write an inequality to represent the number of notebooks (n) and pens (p) they can buy. Identify the feasible region on a graph.

• n + p ≤ 15

• 3n + 2p ≤ 30

• 3n + 2p = 30

• 3n + 2p < 30

6. A local theater has a seating capacity of 300. If tickets for adults cost $10 and tickets for children cost $5, write an inequality to represent the total number of adult (a) and child (c) tickets that can be sold without exceeding capacity. What is the feasible region?

• a + c ≥ 300

• a + c = 300

• a + c ≤ 300

• a + c < 300

7. A charity event aims to raise at least $1,000. If each ticket sold is $20 and each donation is $50, write an inequality to represent the total amount raised. Identify the feasible region for the number of tickets (t) and donations (d).

• 10t + 25d >= 1000; feasible region: t >= 0, d <= 0.

• 20t + 50d >= 1000; feasible region: t >= 0, d >= 0.

• 20t + 50d <= 1000; feasible region: t <= 0, d <= 0.

• 30t + 40d >= 1000; feasible region: t >= 0, d >= 10.

8. A car rental company charges a flat fee of $50 plus $0.20 per mile driven. If a customer wants to spend no more than $100, write an inequality for the number of miles (m) they can drive. What does the graph of this inequality look like?

• m ≤ 200

• m ≤ 300

• m ≤ 150

• m ≤ 250

9. A bakery sells cupcakes for $2 each and cookies for $1 each. If a customer has $40 to spend, write an inequality to represent the number of cupcakes (c) and cookies (k) they can buy. Identify the feasible region on a graph.

• c + k ≤ 40

• c + 2k ≤ 40

• 3c + 2k ≤ 40

• 2c + k ≤ 40

10. A concert venue has a maximum capacity of 500 people. If tickets for VIP seating cost $50 and general admission tickets cost $20, write an inequality to represent the number of VIP (v) and general admission (g) tickets that can be sold. What is the feasible region?

• v + g ≥ 500

• v + g < 500

• v + g ≤ 500

• v + g = 500